Appendix: Valued Fields
In this Appendix we give a brief introduction to the theory of valuations on fields, with special emphasis on Henselian valued fields.
A.I Valuations
Let K be any field.
Definition A.I.!: A subring 0 ~ K is called a valuation ring of K if for all a E K x: a E 0 or a-I EO.
Examples A.I.2: (a) 0 = K (the trivial valuation ring). (b) If:::; is an ordering of K, then
O(Z , :::;) ={a E K I 3n E N, ±a:::; n } is a valuation ring (see (1.1.15)). (c) IfpEN is a prime number, then
Z (p) := { ~ I a, s« Z, p ,r b }
is a valuation ring of Q.
Rernarks A.I.3: (i) A valuation ring 0 of K is a local ring, i.e., 0 has exactly one maximal ideal, mo:= O\OX (Exercise 1.4.8(a)). (ii) O/mo is called the residue field of O.
Let (r,:::;) be an ordered Abelian group (compare Exercise 1.4.4). We shall often append an extra element, denoted by 00, to T; we then extend the ordering to the set r u {(X)} by declaring that r < 00. Moreover, we let l' + 00 = 00 +l' = 00 for all l' E r. 204 Appendix: Valued Fields
X Definition A.1.4: A valuation of K with value group v(K ) is a mapping
v:K~ru{oo} such that for all a, b E K, (i) v(a) = 00 {::} a = OJ (ii) v(ab) = v(a) + v(b)j (iii) v(a + b) ~ min{v(a),v(b)}.
Properties A.1.5: (1) VIKx : K X ~ T is a group homomorphism. Therefore
X 1 v(l) =0 and, for a E K , v(a- ) = -v(a). (2) For all a E K, v(-a) = v(a). Proof: 2v(-a) = v((-a)2) = v(a2) = 2v(a). Hence v(-a) = v(a). Q.E.D.
(3) v(a) =1= v(b) =} v(a + b) = min{v(a),v(b)}. Proof: Without loss of generality, we may assume v(a) < v(b). Suppose v(a + b) =1= miniv(a), v(b)}. Then v(a + b) > v(a), whence
v(a) = v((a + b) - b) ~ min{v(a + b),v(b)} > v(a),
a contradiction. Q.E.D. (4) Ov := {a E K I v(a) ~ O} is a valuation ring. Conversely, to every valuation ring 0 of K there is a valuation v with 0 = Ov (Exer cise 1.4.12).
Example A.1.6: Let pEN be prime. For a E QX, write b a = pr ._, C where b, C, r E Z, and p l a and p lb . Then we define
vp : Q ~ Z u {oo} by vp(a) := rand vp(O) = 00.
This is a valuation, called the p-adic valuation on Q. We have Ov p = Z(p).
Example A.1.7: Let k be a field, let X be a single indeterminate, and let K = k(X). For I.s E k[X] \ {O}, we set
Voo ( i) := deg g - deg f. A.I Valuations 205
Then V oo : k(X) -t Z U {oo} is well-defined, and is a valuation.
Remark A.l.8: If v : K -* r U {oo} is a valuation, then mo. = {a E K I v(a) > O} and 0;: = {x E K I vex) = O}.
Thus VIKx : K X ---+t r has 0;: as kernel; therefore K X /0;: ~ r, and the ordering on T induces the following ordering on KX /0;: :
l aO;: ::::; bO;: ¢> ba- E mo. or aO;: = bO;:.
X Lemma A.l.9: Let 0 be a valuation ring of the field K, and let a E K • Then a E mo ¢> a-I ¢ O. Proof: (=» a E mo implies a-I ¢ 0; otherwise, 1 = aa-l E mo. (¢::) a-I ¢ 0 implies a E 0 (since 0 is a valuation ring) . But a ¢ Ox (otherwise a-I EO), whence a E 0 \ Ox = mo. Q.E.D.
Theorem A .l.ID (Chevalley): Let K be a field, let R ~ K be a subring, and let p ~ R be a prime ideal. Then there exists a valuation ring 0 of K with:
R ~ 0 and mo n R = p.
Proof: Recall the notation R p for the localization of Rat p (4.6.6). Let
E = {(A, I) I n; ~ A ~ K, pRp ~ I c A, A a ring, I a proper ideal of A}.
Then E 01 0, since (Rp,pRp) E E.E may be partially ordered as follows: for all (Aj ,1j ) E E (j = 1,2), we declare
(AI, It) ::::; (A 2 , 12 ) :¢> Al ~ A2 , It ~ 12 •
Each nonempty chain { (A j , Ij ) Ij E J} of such pairs (where J is an arbitrary index set) possesses an upper bound in (E, ::::;), namely,
By Zorn's lemma, E has a maximal element (0, m). Claim: m is a maximal ideal of O. Proof: Otherwise, mel c 0 (with proper inclusions) would imply (0, m) < (0,1) E E. Claim : 0 is local. Proof: Otherwise, (0, m) < (Om, mOm) E E. Claim : 0 is a valuation ring. 206 Appendix: Valued Fields
Proof : Otherwise, there would exist an x E K X such that x,x-1 i O. Then (0, m) maximal => (O[x],mO[x]), (0[x-1], mO[x-1]) i E => 1 E mO[x], 1 E mO[x-1]. n m => 1 = Laixi = LbiX-i, i=O i=O for some ao, . .. ,an, bo, . .. , bm Em. Choose the ai, bi so that n, m are minimal; we may assume m:::; n (otherwise, switch x and X-I). Then
m i bo E m, 0 local => L bix- = 1 - bo E Ox i=1 m => 1 = LCiX-i, i=1 m => z" = L CiXn-i i=1 n n-l m i => 1 = L aixi = L aixi + L Cianxn- , i=O i=O i=1 contradicting the minimality of n, and proving the claim. Therefore, 0 is a valuation ring, m = mo, and
R p ~ 0, m 2 pR p => m n u; 2 pR p => m n R p = pRp (since pRp is maximal in R p)
=> m n R = m n Rp n R = pRp n R = p, Q.E.D.
Definition A.l.ll: Let K2/ K 1 be a field extension, and 0 1 ~ K 1, O2 ~ K2 be valuation rings. O2 is called an extension of 0 1 if O2 n K 1 = 0 1 , We denote this statement by (K1 , 0d ~ (K2 , O2 ),
(i) m02 nKI = mOl' (ii) m02 n 0 1 = mOl'
(iii) 0; n K 1 = O~ , and (iv) 0; nOI = O~.
Proof: For x E K(,
x E m02 ¢} X-I i O2 ¢} X-I i 0 1 ¢} x E mOl (=> (i), (ii)).
O~. (iii): O;nKI = (02\mo2)nKI = (02nKd\(mo2nK1) = 0 1 \mol =
(iv): O;nOI = O;n(KlnOd = (O;nKdnOI = O~nOI = O~. Q.E.D. A.2 Algebraic Extensions 207
Theorem A.1.13: Let KdK 1 be a field extension, and let 01 ~ K 1 be a valuation ring. Then there is an extension O2 of 01 in K 2.
Proof: 01 ~ K 1 ~ K 2 and Chevalley's theorem imply that there exists a ~ ~ valuation ring O2 K 2 with 01 O2 and m02 n 01 = mOl. We must show that 01 = O2 n K 1. (~) follows from 01 ~ O2. To show (;2), suppose x E O2n K 1. Then x fI. 01 implies x =I 0 and X-I E mOl = m02n 01. Hence 1 1 = xx- E mo2 , a contradiction. Q.E.D.
A.2 Algebraic Extensions
Let (K1 , 0d ~ (K2 ,( 2) as in (A.1.11) above. For i = 1,2, o, corresponds to a valuation Vi: K i -* riu{oo} (Exercise 1.4.12(b)). Also, vdKx. : K( -* ri, and ker v, = 0;. Therefore K( /0; ~ rio The composite mapping
x id KXK X/O x rv T"' K l Y 2-* 2 2=.L2
O~, ~ x ~ has kernel 0; n Kt = whence r1 x; /O{ Y K2 /0; r2 , by the homomorphism theorem. Therefore we may regard r1 as a subgroup of r2 with the ordering induced by that of r2 , by Remark A.1.8 and A.1.12(i).
Definition A.2.1: Suppose (K1,0d ~ (K2, ( 2). Then the ramification in dex of this extension is e := e(02/01) := [r2 : r 1], where the T, are as above.
Next, the composite mapping
id - 01 Y O2-* 02/m02 =: K 2
has kernel m02 n 01 = mOl· Thus, K 1 = OI/mol Y 02/m02 = K 2. There fore we may regard K 1 as a subfield of K 2 •
Definition A.2.2: Suppose (K1,0d ~ (K2, ( 2). The residue degree of this extension is f := f(02/01) := [K2 : K 1], where K 2 and K 1 are as above.
Lemma A.2.3: Suppose (K1, (1) ~ (K2, ( 2), and, for i = 1,2, Vi: tc.-« ri U {oo} is the valuation corresponding to Oi. Choose WI, • .• ,WI E O2 and 71"1, •• . , 7I"e E K; so that (1) the residues WI, .•• ,WI E K 2 are linearly independent over K 1, and (2) the values V2 (71"1)' •• . , V2 (71"e) are representatives of distinct cosets of r2/r1 • Then for all aij E K 1, 208 Appendix: Valued Fields
V2 (tt aijW(Trj) = min{ v2(aijW(Trj) 11 :5 i :5 t. 1:5 j :5 e}. (A.2.3 .1) i= l j=l
In particular, the products {Wi1rj I i = 1, .. ., i: j = 1, .. ., e} are linearly independent over K 1 •
Proof: For each j E {I, ... , e}, let r/J(j) be any (e.g., the smallest) E {I, .. ., f} such that V2 (aij) = min{ V2 (akj) 11 ~ k ~ f} E r2 U{00}. Fix j.
Claim: (A.2.3.2)
Proof: We may assume that aq,(j),j ¥= 0, for otherwise a ij = 0 for all i, and (A.2.3.2) would be trivial. Then for each i, a ij / aq,(j),j EO2 n K 1 = 0 1 • Then
the latter follows from the fact that W1 ,• •• , W f E K 2 are linearly independent over K 1 , and one of the coefficients is 1 ¥= O. Therefore
f "" aij L--Wi E Ox2' (A.2.3.3) i=l aq,U),j
ai~ Then V2 (t aijWi) = V2 (aq,u) ,j t a .Wi) i= l i=l q,()) ,J
=V2(aq,U) ,j) + V2 (t a a i~ . Wi) i=l q,(J),J = v2(aq,(j),j) (by (A.2.3.3)).
This proves (A.2.3.2). Returning to the proof of (A.2.3.1), let 1 ~ i.i' ~ e, j ¥= i', and aq,(j),j ¥= o¥= aq,(j') ,j ' . We then claim:
(A.2.3.4)
Otherwise, by (A.2.3.2), A.2 Algebraic Extensions 209
V2(a.p(j),j) + V2(7fj) = v2(a.p(jI) ,j l) + V2(7fj') => V2(7fj) - V2(7fj') = v2(a.p(j') ,j') - v2(a.p(j),j) E r 1 (since all aij E Kd => V2(7fj ) and V2(7fj') represent the same coset in r21r1 , contradicting the assumption, and proving (A.2 .3.4). To conclude the proof of (A.2 .3.1), observe: (~aijWi7fj) V2 = V2(t(t aijWi) 7fj ) t,J J=1 t=1 I = ~inv2 ((~aijWi)7fj) (by (A.2.3.4) and (A.1.5)(3))
I = mjn{V2 (~a ijWi) + V2(7fj)}
= m~n{m~n{v2(a ij)} +V2(7fj)} (by (A.2.3.2)) J t = II).i~ v2(aij7fj) t,J
= min.. v2(atJ' ·wt ·7f·)·J , t,J
the last equation follows from Wi E O2 \ m02 = 0; = kerv2, which in turn follows from Wi ~ 0, which follows from the K1-linear independence of WI, ... ,WI' This proves (A.2.3.1). Finally, to prove the K1-linear independence of the products Wi7fj , observe that for all aij E K 1 ,
o L aijWi7fj i ,j
=> 00=V 2(~::;:ai jWi7fj) =IIf,~nv2(aijWi7fj) (by (A.2.3.1)) t,J => Vi,j, v2(aijWi7fj) = 00 => Vi,j, aijWi7fj = 0 => Vi,j, aij = 0 (by the choice of Wi and 7fj). Q.E.D.
Theorem A.2.4: Suppose (K 1,0d ~ (K2,( 2), and write n = [K2 : KIl, e =e(02/01), and f = f(02l0d· lfn < 00, then e.] < 00 and
ef ~ n.
Proof: In order to exclude the possibility that e or f may be infinite, we begin by considering any e', f' < 00 (e'.l' E N) such that e' ~ e and f' ~ t, it then suffices to show that e'f' ~ n. The latter follows from the fact that the 210 Appendix: Valued Fields e'f' products W(lT"j with i ~ f' and j ~ e' considered in (A.2.3) were shown to be linearly independent over K 1 • Q.E.D.
Lemma A.2.5: Suppose 0 1, ... , On are valuation rings of a field K. Let R := n~=l O, and Pi := R n mo•. Then for 1 ~ i ~ n, O, = Rpi •
~ ~ Proof: R Pi O, is clear. To prove 0 1 Rpll let a E 0 1, and let II = {i I a E Oi}. Set m, = mOi and (}:i = a + m, E O;jmi for each i E h. Choose a prime number pEN so that (1) p> charO;jmi for all i E II, and (2) (}:i is not a primitive pth root of 1, for all i Ell, 1 Set b = 1 + a + ...+ aP- • Then
=? b= 1 + ... + 1 = p "# 0 in O;jmi, - 1 - (}:~ =? b = -1--l "# 0 in O;jmi' - (}:i
Thus, either way, b E 0 t for all i Ell' For i E {I, . . . ,n} \ II, aft Oi, whence a-I E mi. Hence
~ 1 + a-I + ...+ a- (p-1) E 0 l , implying
~ = a-(p-1) . 1 0 and b I + a-I + ... + a-(p-1) E i , a a 1 1 1 ( 1) E Vi- b= aP- 1 + a- + ... + a- P-
Thus for all i = 1, ... , n, lib, alb E Oi, i.e., lib, alb E R. From lib ft m1 n R = PI follows alb a = lib E R p1· Q.E.D.
Theorem A.2.6: With the assumptions and notations of Lemma A .2.5, sup pose that o, g OJ for all i"# j. Then (i) for all i "# j, Pi g Pj, (ii) for all i = 1, . . . , n , Pi is a maximal ideal of R, and (iii) for each n -tuple (a1, ' .. , an) E 0 1 X •.• X On , there exists an a E R with a - ai E mi.
~ ~ Proof: (i) If Pi Pj then OJ = R pi RPi = o.. by (A.2.5). (ii) Let a be an ideal of R with a "# R; by (i), it suffices to show that a lies in some Pi. Otherwise, for each i = 1, ... , n, pick ai E a \ Pi. For each i "# i. use (i) to pick bij E Pi \ Pj. Then A.2 Algebraic Extensions 211
n Cj := IIbi j E n(Pi \ pj) , and d := L ajcj f/:. Pi· i#j i # j j=l
Then d- l E n7=l O, = R implies 1 = dd-l E a, i.e., a = R, contradiction. (iii) For i i i. Pi + Pj = R, using (ii) and (i). Therefore the canonical map R ~ Rlpl X ••• x RIPn is surjective (Chinese Remainder Theorem). Since for each i,
Rlpi ~ RpJpiRpi (by (ii) and (3.6.8)) =Oi/mi (by (A.2.5)) ,
R ~ Ot/ml x .. . X Onlmn is surjective. Q.E.D .
Lemma A.2.7: Suppose LIK is an algebraic extension of fields, 0 is a valuation ring of K, and 0' and 0" are valuation rings extending 0 to L . Then if 0' ~ 0", then 0' = 0".
Proof: Let us first note the easily checked but fundamental equivalence
0' ~ 0" ¢:} m" ~ m', (A.2.7.1) which holds for all valuation rings 0' and 0" of L with maximal ideals m' and m", respectively. Returning to the proof of the lemma, it clearly suffices to show the statement of the lemma for finite extensions LIK. By assumption we have O1m ~ 0'1m" ~ 0"1m" (the first inclusion using m = K n m"). Since [0"1m" : Olm] ~ [L : K] < 00 by (A.2.4), the integral domain 0'1m" is a finite-dimensional Olm-vector space. Thus it is a field, whence m" is a maximal ideal in 0'. Therefore m" = m', whence 0" = 0'. Q.E.D.
Theorem A.2.8: Suppose LIK is a finite Galois extension of fields, with G = Gal( LIK). Suppose 0 is a valuation ring of K, and 0' and 0" are valuation rings in L extending O. Then 0' and 0" are conjugate, i.e., there exists a E G with aO' = 0".
Proof: Let H' ={a E G laO' =O'} and H" ={rEG IrO" = 0" }. Then H' and H" are subgroups of G. Write G as disjoint unions of cosets of H' and H", respectively:
n m G= UH'a;l and G = H"r-:- 1 (A.2.8.1) U J' i=l j=l 212 Appendix: Valued Fields for suitable ai, Tj E G. Suppose, for the sake of contradiction, that
for all i,j, O'iO' ~ TjO" and TjO" ~ O'iO' . (A.2.8.2)
Set n m R = nO'iO' n nTjO". (A.2.8.3) i=1 j=1 None of the n + m valuation rings in (A.2.8.3) contain any of the others, by (A.2.8.2) and the fact that for all i,i' E {I, . .. ,n},
, C 0' -1 0' 0' - 1 H' .., O'i 0 _ ai' ~ O'i O'i' = ~ O'i ai' E ~ t = t , (A.2.8.4) (A.2.7) and analogously for H". (A.2.6)(iii) then gives an a E R with
a-I E aim', for i = 1, , n, and a - 0 E Tjm", for j = 1, , m.
From (A.2.8.1) it then follows that
O'(a) Em' + 1 for all a E G, and T(a) E m" for all T E G.
Then NL/K(a) = II O'(a) E (m' + 1) n K = m + 1, and O'EG NL/K(a) = II T(a) E m" n K = m, TEG contradiction. So (A.2.8.2) is false; i.e., for some i,j, we get O'iO' ~ TjO" or TjO" ~ O'iO'. Thus O'iO' = TjO" (by (A.2.7)). Hence 0" = Tj-10'iO'. Q.E.D .
Conjugation Theorem A.2.9: Suppose LjK is an arbitrary Galois exten sion of fields, 0 is a valuation ring of K, and 0' and 0" are valuation rings in L extending O. Then there exists a E Gal(LjK) with 0'0' = 0".
Proof: Consider the set of ordered pairs (K1, ad, where K 1 is an intermediate extension of L jK, O~ = 0'nK 1, O~ = 0"nK 1 , and 0'1 is an automorphism of KdK with 0'10~ = O~ . We endow the set of such ordered pairs (Ki ,O'i) with the partial ordering
By Zorn's lemma there exists a maximal such pair (Km,am) with K ~ Km ~ L and O'm(O~) = O~ , where O~ := 0' n Km and O~ := 0" n Km. A.3 Henselian Fields 213 K I aEL 0' 0" /\ N 0* -0**a--+- am 0" n N \/ I , O~ K m 0'm am
K
It suffices to show that K m = L. Otherwise, we could pick a E L\Km . Let f = Irr(a, K), and let N be the splitting field of f over K m. We extend am to an automorphism (still denoted by am) of the algebraic closure K of K. Then am(L) = Land am(N) = N. Let 0* := 0' n Nand 0** := a:;;/(O" n N). Then 0* n tc; = 0** n tc; = O:n. Application of (A.2.8) to 0* and 0** gives a a E Gal(NIK m) with 0** = «O", Then
am 0 a(O' n N) = amO** = 0" n N.
Thus (N, am 0 a) > (Km,am), contradicting the maximality of (Km,am) ' Q.E.D.
A.3 Henselian Fields
Suppose LIK is a Galois extension of fields with G := Gal(LIK), 0 is a valuation ring of K, and 0' is an extension of 0 to L.
Definitions A.3.l: Z(O') := {T E G I TO' = O'} is called the decomposi tion group of 0'10. The fixed field Kz of Z(O') is called the decomposition field of 0'10. If L := KS is the separable closure of K,l then (Kz, 0' n K z) is called the Henselian closure or Henselization of (K, 0); cr. (A.3.11) below. Case 1: LI K finite Let H = Z(O') and m = [G : H], and write G as the disjoint union of cosets of H:
8 1 Note : K = {a E K I a is separable over K}, where K denotes some fixed algebraic closure of K. 214 Appendix: Valued Fields
G = 0'1-1 H U ... U 0'-1Hm , (A•.3 1• 1) for suitable a, E G; without loss of generality, we may take 0'1 = id. Then 0', 0'21C)I , .. .,0';;/0' are extensions of ° to L; there are no others, by (A.2.8), and there are no repetitions in this list; d. (A.2.8.4). For i = 1,oo.,n, write K~l = O'i(Kz)j then K~) = 0'1 (Kz) = K z. 1 O[i] := O'i (0') n K z is a valuation ring of K z. Note that
Here are two diagrams of our situation:
L (L,O') /~
(K~) ,o'nK~)) 00. (Kkm],O'nKkm)) ~/ (K,O) K ° Lemma A.3.2: 0' is the only extension of 0[1) to L.
Proof : If O'i10' n Kz = 0[1] = 0' n K z , then by (A.2.8) there would exist l T E H with TO'i10' = 0', whence TO'i E H, i.e., O'i = id. Q.E.D.
Lemma A.3.3: The residue degree (A.2.2) f(O[l]/O) = 1.
Proof: Let m m 1 R = nO[i] = nO'i (0' n K~)) C; K z. i=l i=l Let 0: E 0[1]. We must show that there exists an a E ° with 0: - a E m[l]. For this, choose {3 E R with
{3 - 0: E m[l] and, (A.3.3.1) for i = 2, .. . ,m, {3 E m[i] = O'i1m' n Kzj (A.3.3.2) such a (3 exists, by (A.2.6)(iii). Set
m a = LO'i({3). i=l A.3 Henselian Fields 215
Then a E K, since a is invariant under G; this is because, for any a E G, the elements aai , ..., aam will be another system of coset-representatives of G/H, which will map /3 to the same images (after a permutation) to which the al, .. ., am mapped /3, using the fact that /3 E R ~ K z- Therefore
m a -/3 = Lai(/3) Em' nKz = m[l] (using a,/3 E Kz and (A.3.3.2)) i= 2 => a - a = (a - /3) + (/3 - a) E mll] (using (A.3.3.1)). Q.E.D.
Lemma A.3.4: The ramification index (A .2.1) e(Oll]/O) = 1.
Proof: Let a E K;. We must show that there exists an a E KX with v[l](a) = v[l](a). For this, choose /3 E R with
1 - /3 E m[l] and , for i = 2, .. . , m /3 E mli); such a /3 exists, by (A.2.6)(iii). Then
V[l) (/3) = 0 and, for i = 2, , m, vIi)(/3) > O. I.e., v' (/3) =0 and,for i=2, ,m, v' (ai (/3)) >0.
It is therefore possible to choose an n E Z such that
Letting a' = /3n a , we get v'(a') :j; v'(ai(a')), for i = 2, . . . , m. Set
w = {i I v'(ai(a')) < v'(a')}, w = #W, and rw = L IIai(a'). I~{l,... ,m} iEI #I=w Then
(since all other summands in r w v'(rw) = v' ( ai(a')) II have higher value), and iEW v'(rw+d = v' (a'II ai(a')) ' iEW Then a:= rwH E K rw and v'(a) = v[lJ(a') = vllJ(a). Q.E.D. 216 Appendix: Valued Fields
Case 2: LI K arbitrary Let L1/K be a finite Galois extension, and let L1 ~ L. Then by (A.2.9) we find, for G1 = Gal(L1/K) and Z1(0') = {a E G1 Ia(O'nL1) = O'nL1}, that Inv Z1(0') = L1n Inv Z(O') = L1n Kz (where, for any subgroup H ~ G, Inv H denotes the subfield of L fixed by H).
0' L /\ Oz := 0' nKz Kz L1 \ / O'nKznL1 KznL1
o K
Corollary A.3.5: Oz = 0' n K z has exactly one extension to L, namely, 0'.
Proof: Let 0" be a second such extension, and suppose we can find an a E 0' \ 0". Let L1IK be a finite Galois extension with a E L1 ~ L. Then
0' n K z n L 1 = 0" n K z n L 1 and
a E (0' n L 1 ) \ (0" n Lt), contradicting (A.3.2). Q.E.D.
Corollary A.3.6: The residue degree (A.2.2) j(OzIO) = 1.
Proof: Let a E Oe with a + mz ¢ O/m. Choose a finite Galois extension L1/K with a E L1 ~ L. We then contradict (A.3.3). Q.E.D.
Corollary A.3.7: The ramification index (A .2.1) e(OzIO) = 1.
Proof: Let a E K; with vz(a) ¢ v(K). Choose a finite Galois extension L1/K with a E L1 ~ L. We then contradict (A.3.4). Q.E.D.
Definition A.3.8: A valued field (K,O) is called Henselian if 0 has a unique extension to the separable closure KS of K. A.3 Henselian Fields 217
Note that (K,O) with the trivial valuation 0 = K is Henselian, since by (A.2.7) the trivial valuation extends only to the trivial one on any algebraic extension of K.
Definition A.3.9: Suppose K'/ K is a field extension, and the valuation ring 0 ~ K extends to 0' on K'. The extension 0'/0 is called immediate if e(O' /0) = f(O'/0) = 1.
Applying (A.3.5-7) to L = K8, we get :
Theorem A.3.IO: The Henselization (K', 0') of (K, 0) (A .3.1) is an im mediate extension, and (K', 0') is Henselian.
Remark A.3.11: The Henselization (Kz,Oz) is determined by the exten sion of 0 to 0' on K8 . Different extensions are conjugate. (Kz, Oz) is there fore determined only up to isomorphism as a valued field over (K,O).
Theorem A.3.I2: The Henselization (Kz,Oz) of (K, 0) has the following characterization: (1) (Kz,Oz) is Henselian, and (2) if (K,O) ~ (KI,OI) and (KI,Od is Henselian, then there exists a uniquely determined embedding A: (Kz,Oz) -+ (KI,OI) with AIK = id .
Proof of (A.3.12): Corollary A.3.5 shows (1). We have to show that (Kz, Oz) satisfies (2). Since every relatively separably closed subfield of (KI , 0d is also Hense lian with respect to the induced valuation with K I (see (A.3.14) below) , it suffices to consider the case in which KdK in (2) is separable. 8 Let 0 be the uniquely determined extension of 0 1 on K8. Then Ko := 8, Inv Z(08) ~ K I, for in case a E Gal(K8/ K I), then a(08) = 0 so a E Z(08). Therefore 8 Inv Z(08) ~ InvGal(K /Kd = K 1 •
If Kz = Inv Z(O'), then there is a A E Gal(K8/ K) with A(O') = 0 8, whence
A: Kz -+ Inv Z(08) = K o and Z(08) = AZ(O')A-I . Also, A is uniquely determined: for suppose 8 p: Kz -+ Ks, PIK = id, and p(Oz) = 0 n K o =: 0 0 •
8 Extend p to K • Then Kz n A- I(08) = Oz = Kz n p-I(08), hence 0' = r 1 (0 8) = p-I(08) = p-lA(O'). 218 Appendix: Valued Fields
1 Therefore p- ).. E Z(O') and therewith )..IKz = plKz . Q.E.D.
The next theorem will give some equivalent conditions for a valued field (K,O) to be Henselian. All equivalent conditions will talk about (zeros of) polynomials f E O[X] in one variable. There are, of course, many such equiv alents known. Here we concentrate on those used in the course of this book. Observing that (5) =} (1) uses only a separable polynomial , it is easy to see that in the conditions (3) to (6) it suffices to consider only separable polynomials from O[X] (where separable means without multiple zeros). Here it is convenient to mention and to use an elementary result that is proved in Section A.6 in more generality: Suppose v is the valuation corresponding to O. Then the definition
(for ai E K), and w(f/g) = w(f) - w(g) (for I,e E K[X] \ {O}) yields a valuation w on K(X), by (A.6.3). This extension of v to K(X) is called the Gauss extension. The property w(fg) = w(f) + w(g) will be used from now on in the following way. Let us call a polynomial f E O[X] primitive if w(f) = 0, i.e., if at least one coefficient of f is a unit in O. Now dearly the product of primitive polynomials from O[X] is again primitive, and if a primitive polynomial f E O[X] has a factorization f = gh in K[X], then it also has a factorization f = glh1 in O[X] with gl and li: both primitive, and being constant multiples of f and g, respectively.
Theorem A.3.13 ("Hensel's Lemma"): For a valued field (K,O) with residue field K and residue homomorphism a H a, the following are equiva lent: (1) (K,O) is Henselian. (2) Let I, g, h E O[X], where f has only separable irreducible factors, ] = gTi =I 0, and (g,Ti) = 1. Then there exist gl , b, E O[X] with f = ssb« , gl = g, hI = t; and degg1 = degg. (3) For each f E O[X] and a E 0 with ](a) = 0 and 7' (a) =I 0, there exists an 0: E 0 with f(o:) = 0 and Q =a. (4) For each f E O[X] and a E 0 with v(f(a)) > 2v(f'(a)), there exists an 0: EO with f(o:) = 0 and v(a - 0:) > v(f'(a)). n (5) Every polynomial x + an_1xn-1 + ... + aD E O[X] with an-1 ¢ m and an-2,..., aD E m has a zero in K. (6) Every polynomial X" + X n-1 + an_2xn-2 + ... + aD E O[X] with an-2, ... , ao E m has a zero in K.
Proof: Let L be the splitting field of f over K.
(1) =} (2): Let 0' be the unique extension of 0 to L (using (1), (A.3.8), and (A.1.13)). Let f := anxn +... + aD E O[X]. Since] =I 0, f is primitive. In L we have A.3 Henselian Fields 219
n I =IICBiX - Qi), o; Qi E 0', f3i i 0, i=l with min{v(f3i), V(Qin = 0, i.e., (f3i,Qi) = 1. We may suppose that
m g=lII(f3iX -ai), f.,f3i E (O,)x i=l (possibly after re-numbering the factors) . Set
Such a c exists because f. IT:l f3i is the leading coefficient of 9 E K[X]. Then gl = 9 and deggl = degg = m. Now set hI = I jgl' Then
We shall show that (each coefficient of) gl is invariant under all a E Gal(LjK); it will then follow that gl,h1 E O[X]. From a(Ql= 0' follows a(m') = m'. Thus a defines a mapping (j : L -t L by a I--t a(a), which is an automorphism ofLjK. From (g, h) = 1 it follows that for each i E {I, .. . ,m} there exists j E {I, ... ,m} such that a(;;) = ;~.
Thus a permutes the zeros of gl, whence the coefficients of gl lie in K, and therewith gl E O[X]. (2) ~ (3): First suppose I is separable. Set g(X) = X - a and li =tjg E K[X]. Then 7 = gli and (g, li) = 1, since I'(a) i O. There exist gl, hI E O[X] with I = glh1, gl = 9 = X - a, and deggl = 1 = degg, by (2). It then follows that gl = e(X - b) with e E Ox and b EO. Then e = 1, I(b) = 0, and b = a. Now let I be inseparable, and write I = fIh, with fI, h E O[X], where fI is the product of the separable irreducible factors of I, and h is the product of the inseparable irreducible factors of I. Then heX) = h(XP), for some hE O[X], where p = char K = charOjm > O. From 7(a) = 0 and I'(a) i 0 follows fICa) = 0 and I{(a) i 0 (since p > 1). Then the previous paragraph implies that fI has a zero Q E K with a =a, so I has one, too . (3) ~ (4): I(a - X) = I(a) - f'(a)X + X 2g(X), for some g E O[X]. Writing X = f'(a)Y, and observing that vU'(a)) i 00 and hence f'(a) i 0, we get 220 Appendix: Valued Fields
f(a - f'(a)Y) f(a) 2 f'(a)2 = f'(a)2 - Y + Y h(Y) = : it(Y). Then it E O[Y], since v(f(a)) > v(f'(a)2). Now it = Y(Yh(Y) -1), which has the simple zero 0 in the residue field. Therefore it has a zero y E m, by (3). Then f has the zero a := a - f'(a)y E O. Since y Em, v(a - a) > v(f'(a)). (4) :::} (5): Let f = X" + an_1xn-1 + ...+ ao as in (5). Then n-1 n-1 7=x»+ an_1X =X (X + an-d.
Then -an - 1 (#0) is a simple zero of]. In particular, v(f(-an-d) > 0 = v(f'(-an-d). Then f has a zero in 0, by (4). (5) :::} (6): Trivial. (6) :::} (5): Suppose f(X) = xn + an_1Xn-1+ ...+ ao with an-1 E Ox and an-2,"" ao E m. Replace X by an-1Y and divide by a~_l; we obtain g(Y) = y n + y n- 1+ a;-2 yn-2 + ... + ~o . an - 1 an - 1 Apply (5) to g(Y) to obtain a zero y E K of g. Then x := an-1Y is a zero of f· (5) :::} (1): Suppose (K,O) were not Henselian. Then there would be a finite Galois extension L / K in which 0 extends to 0' and 0", with 0' # 0". It followsthat Z(O') # Gal(L/K), since by (A.2.8), 0' and 0" are conjugate over K. Hence m ~ 2 in (A.3.1.1). As in the proof of (A.3.3), and writing ,ali] = a, (,a) , there exists ,a E R = n~l O[i] with ,a[1] - 1 E m' and , for i = 2, . . . ,m, ,a[i] Em'. Then
m m m 1 f := II(X - ,a[i]) = X + am _ 1X - + ...+ ao E O[X], i= l
-am-1 = L ,ali] == 1 mod m, am-2 == .. . == ao == 0 mod m. Then f has a zero in K, by (5). Hence ,a E K and thus ,ali] = ,ali] for all i,j. This contradicts ,a[l) == 1 mod m and ,a[2) == 0 mod m. (Note: f is separable.) Q.E.D.
Corollary A.3.14: Let (K',O') be Henselian, K ~ K', and 0 = KnO'. If K is relatively separably closed in K', then (K,O) is Henselian.
Proof: We use (1) :::} (5) and (5) :::} (1) of (A.3.13): Let n n-1 f = X + an_1X + ...+ ao E O[X] be separable, an-1 ~ m, and an-2 ,"" ao E m. Then f has a zero in K', hence also in K. Q.E.D . A.3 Henselian Fields 221
Definition A.3.l5: A valued field (K,O) is called algebraically maximal if it admits no proper, algebraic, immediate extension (K', 0').
Note that K with the trivial valuation is algebraically maximal.
Definition A.3.l6: A valued field (K,O) is called finitely ramified if either char K = 0, or char K = p > 0 and there are only finitely many values between 0 and v(P) .
Note that (K,O) with 0 = K is finitely ramified, and that if (K,O) is finitely ramified and 0 is nontrivial, then char K = O.In fact, if char K = p > 0, then there are infinitely many elements between 0 and v(P) = v(O) = 00 in the value group.
Examples A.3.l7: (1) Let:::; be an ordering of K, and let 0 = O(Z,:::;) (A.1.2)(b). Then K is ordered, whence char K = O. (2) If To ~ Z and char K = 0, then (K,O) is finitely ramified.
Remark A.3.l8: Suppose (K,O) is finitely ramified. Then for every n E Z\ {O}, there are only finitely many values between 0 and v(n) . To see this, we consider the two cases, char K = p and char K = O. If char K = p, write n = pes with p ,.r Sj then v(n) = ev(p), so that there are e times as many values between 0 and v(n) as between 0 and v(P) (approximately). Now suppose char K = O. Since in this case char K = 0, Q ~ K, and mo n Q = (0) ~ 0, so that for all T E Q, r = T. Since char K = 0, for all n EZ\ {O}, fi f; 0, whence v(n) = O. Thus also in this case, there are only finitely many values between 0 and v(n).
Theorem A.3.l9: Suppose (K,O) is finitely ramified. Then (K,O) is Henselian if and only if (K, 0) is algebraically maximal.
Proof: (~) Let (K,O) be algebraically maximal. Then (K,O) is Henselian, since the Henselization is an algebraic, immediate extension. (=» Let (K', 0') ;2 (K,O) be a proper, algebraic, immediate extension. Then clearly 0 f; K, and thus char K = O. Let a E K' \ K. Without loss of generality, suppose K' j K is finite, and let L be the normal closure of K'j K . Then 0 extends uniquely to L. In particular, this extension also extends 0' from K' to L. Now
v({3) = v«(1({3)), for all {3 ELand (1 E G := Gal(LjK).2 (A.3 .19.1)
Let a[l) = a, a[2] , . .. , a[n] be the conjugates of a. Then
2 This follows from the fact that eriK = id or that the order of a is finite (cf. Exercise A.7.4(iii)). 222 Appendix: Valued Fields
1 n a := - L ali] E K. n i = 1
We have a - a =/= 0 and v(a - a) = "I E v(K') = v(K). Since 0'/0 is immediate, there exists c E K with v(c) = "I, whence
v(a: a) = O.
In addition, there exists a d E K with
a - a ) v ( -c- -d > O.
It therefore follows that
v(a - ~) > v(c) = v(a - a). s« K Through finitely many repetitions we obtain abE K with
v(a - b) > v(a - a) + v(n), (A.3.19.2) using (A.3.18). Then, in particular,
v(a - b) = v((a - b) - (a - a)) = v(a - a). (A.3.19.3) Summarizing, we get v(n) + v(a - b) = v(n(a - b» =v(~(ali]-b))
I ~ v(a - b) (by (A.3.19.1)) > v(a - a) + v(n) (by (A.3.19.2» = v(a - b) + v(n), (by (A.3.19.3» contradiction. Q.E .D.
Corollary A .3.20: If (K,O) is finitely ramified, then the Henselization of (K,O) is characterized as the algebraically maximal extension.
Proof: Let (K', 0') be algebraically maximal over (K,O). Then (K', 0') is Henselian. Therefore the Henselization (K", 0") of (K,O) is contained in (K', 0'), by (A.3.12)(2). Since the Henselization (K",O") is an immediate extension of (K,O) (A.3.10), it, too, is finitely ramified. Thus (K", 0"), being Henselian, is al gebraically maximal, by (A.3.18). Therefore K" = K'. Q.E.D. A.4 Complete Fields 223 A.4 Complete Fields
Every valuation v : K -t r u {oo} on a field K induces a Hausdorff topology on K that turns K into a topological field, as follows. For each a E K, the sets U-y(a) = { x E K I v(x - a) > , }, , E r, form a basis of open neighborhoods of a: (1) a E U-y(a);
(2) U-Yl (a) nU-Y2(a) =Um ax h l ,-Y2} (a); (3) bE U-y(a) , b f. a, v(b - a) = " >, imply U-y' (b) ~ U-y(a), since v(x - b) > " = v(b - a) implies v(x - a) = v((x - b) + (b - a)) = v(b - a) = "(' > , . Consequences A.4.1: (i) v trivial {:} Ov = K {:} r = {O} {:} U-y(a) = {a} {:} the indu ced topology is discrete.
From now on, v is nontrivial. (ii) The sets {x Iv(x - a) 2: ,}, {x Iv(x - 0,) ~ ,}, and {x Iv(x - a) = , } are open. For since v(x - b) > v(b - a) => v(x - a) = v(b - a), we have, for example,
{x I v(x - a) s ,} = U UV(b - a)(b) . v(b-a )~ -y
Therewith are all of these sets (and of course also U-y(a)) both open and closed. This applies, for example, to 0 = {x I v(x) 2: O} and m = {x I v(x) > O}. (iii) The field operations are continuous with respect to this topology. For example, v(x + y) 2: min{v(x), v(y)} implies U-y(xo) + U-y(Yo) ~ U-y(xo + Yo).
Definition A.4.2: A valuation v : K -tt T u {oo} is said to be of rank 1 if r is Archimedean ordered (i.e., embeddable in (JR,+, <)).
Let v : K -t lR U {oo} (not necessarily surjective). Then the equation
Ix - ylv := e- v(x- y) (with e-oo := 0) defines a metric on K: 224 Appendix: Valued Fields
(1) Ix - ylv = 0 {:} Vex - y) = 00 {:} x = y. (2) Ix - ylv = Iy - xlv. (3) Ix-ylv=l(x- z)+(z-y)lv ~ max{lx - zlv, Iz- ylv} ~ Ix - zlv + Iz - ylv.
Let (Xn)nEN be a sequence in K, and let a E K. Then
lim X n = a {:} (Vf. > 0) (3no E N) (Vn ~ no) la - xnl < f. n-+oo {:} (V, E F) (3no E N) (Vn ~ no) v(a - x n ) > , .
Observe: , -+ 00 {:} e-"( -+ O. And
(Xn)nEN is a Cauchy sequence {:} V, 3no (Vn,n' ~ no) v(xn - xn') > ,. Definition A.4.3: A rank-I valued field (K, v) is called complete if every Cauchy sequence in K converges.
Example A.4.4: Consider (Q, vp ), where pEN is prime, and vp : Q -+ z U {oo} is the valuation determined uniquely by the following requirement: for a,b EZ and mEN,
vp(a - b) 2: m {:} pmla - b (AAA.I) {:} a == b (mod pm).
Note that Ovp = Z(p) from Example A.1.2(c) . For this valuation we find lim pn = O. n-+oo
Fix m E Z. For all i ~ m, let ai E {O,. .. ,p -I}, and for all n ~ m, let
(AA.4.2)
Claim A.4.5: The sequence (xn)n2: m is a Cauchy sequence.
Proof: Given, EN, let no = ,. Then
, _ n'+l + + n n > n ~ no =} Xn - Xn' - an'+l P ... anP =} vp(xn - xn') ~ n' + 1> ,. Q.E.D . We write 00
lim X n aipi; n-+oo = '"'L...J i=m AA Complete Fields 225
while such a limit need not exist in Q, we shall soon see that such a limit
always exists in the completion of Q with respect to I. lup • First we prove
Claim A.4.6: Every r EQ is such a limit.
Proof : Without loss of generality, let r :I 0 and m = vp(r) . Then vp(rp-m) = 0, so there exists an am E lFp \ {O} = {I, . . . ,p -I} with v(rp-m - am) > O. Then v(r - ampm) > v(pm) = m, '---v----" rm+l and therewith v(rm+d 2: m + 1. Now suppose , using induction on i 2: m, that we have found am, . . . ,ai-l E lFp such that, letting
ri := r - amPm _...- ai-IPi - I , j := v(ri) 2: i. We shall define ai, ... , aj E lFp such that
(AA.6.1) this will then show that
as desired. To prove (AA.6.1), first observe that v(riP-j) = O. Thus there exists an aj E lFp \ {O} such that v(riP-j - aj) > O. For v = i, . .. , j - 1, define av = O. Then
Q.E.D.
Observe: The ai and m are uniquely determined by r.
Every metric space has a completion. In this completion, every Cauchy sequence converges. For valued fields, even more holds. In (AA.ll) we shall show that every valued field (K, v) admits a valued field extension (K, v) in which every Cauchy sequence'' converges and in which K is dense. This extension will be unique up to valuation isomorphism . It is called the com pletion of (K,v). Assuming this for the moment, it is not difficult to see that (AA.6) holds even for the completion Q, := (Q,v;,) of (Q,vp ); i.e., every element r of Qp may be written as a series of the form
3 The notion of a Cauchy sequence will actually be generalized in that case (cf. (AA.9)). For rank-l valuations, however, it reduces to the classical notion used above. 226 Appendix: Valued Fields
00 r = L aipi, ai E {O, 1, ... ,p - I}, mE Z. i=m If am ::j:. 0, one sets
This defines the valuation v;, of Qp. We have
v;, :ij -+ ZU {oo}, with residue field IFp. Qp is called the field of p-adic numbers. The extension (Q, vp ) ~ (ij,v;,) is immediate.
Example A.4.7: Consider (k(X), vp), where k is a field, p = X - e (for some c E k), and vp is defined by analogy with (AAA.l). As in (AA.5), the sequence analogous to (AAA.2) is Cauchy; we now write
00 lim X n = L ai(X - c)i, n-too i=m which will always exist in the completion of k(X) with respect to I·Ivp (see below). Every f E k(X) is such a limit ; i.e.,
00 f = L ai(X - e)i, for some ai E k, m E Z, i=m by analogy with (AA.6) . We call arbitrary expressions of the form
00 L ai(X - e)i (ai E k, m E Z) i=m formal Laurent series about cover k; we denote the field of all such series by k((X - e)); this field is the completion of (k(X),vp). We extend vp to a valuation on k((X - e)) by defining
in case am ::j:. 0; then
vp : k((X - c)) -+ ZU {oo}, with residue field k. Thus the extension (k(X),vp) C (k((X - c)),vp) is immediate. A.4 Complete Fields 227
Theorem AA.8: II the field K is complete with respect to a rank-l valuation v : K --t IR U{00}, then K is Henselian.
Proof: We prove property (4) of (A.3.13). So let I E O[X], ao E 0, and v(f(ao)) > 2v(f'(ao)). We must find an a EO with I(a) = 0 and v(ao - a) > v(f'(ao)).
Let eo =!'(ao), and choose f > 0 so that
Then I(ao) = eazo, where Zo E K and v(zo) 2.: f. Set al := ao - eozo. Then using Taylor's formula and ao,eo, Zo E 0,
I(al) = I(ao - eozo) = I(ao) - eozo!,(ao) + e~z5a (for some a EO) =e~zo - e~zo + e~z5a = e~z5a.
Hence v(f(ad) 2.: v(ea) + 2f and
!,(ad = !'(ao - eozo) = !'(ao) - eozob (for some bE 0) = eo(l - zob) =: el .
Then v(el) = v(eo) and I(ad = erZl' where Zl E K and v(zd 2.: 2f. We repeat this argumentation with el for eo and with a2 = al - el Zl. It follows that !'(a2) = e2 for some e2 with v(e2) = v(eo), and l(a2) = e~z2 for some Z2 E K with V(Z2) 2.: 4f. Iteration leads to with !'(an+l) = en+! and v(en+d = v(eo), and with I(an+!) = e;+!Zn+l n n and v(zn+d 2.: 2 +! f . The sequence (an)nEN is Cauchy, since 2 f --t 00. Indeed, for m ::; n,
Let a = n-toolim an' From the continuity of I (AA.l)(iii) and v(f(an)) = v(ea) + v(zn) --t 00 follows I(a) = I(n-toolim an) = n-toolim I(an) = O. Furthermore, v(f'(a)) = v(eo) < v(a - ao), 228 Appendix: Valued Fields since v(a n - ao) = v((an - an-i) + (an-l - an- 2) + ... + (al - ao)) :2: min v(a v - av-d :2: v(eo) + e, v whence v(a - ao) :2: v(eo) + € > v(eo) = vU'(a)), as required. Q.E.D .
Now let v : K -» r u {oo} and let r be an arbitrary, ordered, Abelian group.
Definition A.4.9: Let K. be the smallest cardinal number serving as the index set of a sequence "Iv (v < K., "Iv E r) that is "cofinal" in T (i.e., to each 8 E T there exists a v < K. with 8 < "Iv) . The cardinal K. is called the cofinality of r.
We consider sequences (aV)V a = lim av :¢:> (V"I E r) (3vo < K.) (V v :2: vo) v(a - av) > "I; V Definition A.4.10: (K, v) is called complete if every Cauchy sequence in K converges. Theorem AA.ll: Every valued field (K,v) possesses one and (up to valua tion isomorphism) only one valuation extension (R, v) that is complete and in which K is dense. ((R,v) is called the completion of (K,v).) Proof: See Exercises (A.7.6), (A.7.7), and (A.7.8). Definition/Notation AA.12: (K, v) is called relatively complete if K is relatively separably closed in R. For a valued field (K,v), we write (Ka,va) for the relative separable closure of Kin R, where va = VIKa. Remark AA.13: (K,v) ~ (Ka,va) ~ (R,v) are immediate extensions. Proof : Given 0: E R \ {O}, by the density of K in R, choose a E K with v(o: - a) > v(o:). Then v(a) = v((o: - a) - 0:) = v(o:). Then f =-!. For v(o:) = 0, it follows, moreover, that v(o: - a) > 0, whence a = a, i.e., R = K. Q.E .D. AA Complete Fields 229 Theorem A.4.14: A valued field (K, v) is relatively complete if and only if every separable polynomial f E K[X] that comes arbitrarily close to 0 over K (i.e., 0 is in the closure of f(K)) has a zero in K. Proof: (<=) Suppose (K,v) i- (Ka,va) and a E Ka \ K. Note that f := Irr(a, K) comes arbitrarily close to 0 over K, since K comes arbitrarily close to a. Yet f has no zero in K. (=» Suppose K = K", and f E K[X] comes arbitrarily close to 0 over K, and is separable over K, with degf = : d. We may assume f is monic. Let "Iv (v < "') be a cofinal sequence in r. Then for each v < '" there exists an av E K with v(J(av)) > d'Yv. We extend v from K to (K)s. In (K)S we have The value group I" of (K)S is contained in the divisible hull of r, using ef ~ n. In particular, ("{V)V d d'Yv < v(J(av)) = Lv(av - ai) i= l it follows that for at least one i, Therefore there exists an i and a subsequence of av that converges to ai. Since K is complete, a i E K. Since K is relatively complete, o, even lies in K. Q.E .D. Theorem A.4.15: If (K, v) is Henselian, then so is (K, v) and hence also (Ka, va). Proof: Using (A.3.13)(5), let f = xn + Cxn-l + ... + ao E K[X] with v(C) = 0 and, for i ~ n - 2, v(lii) > O. We must show that f comes arbitrarily close to 0 (A.4.14). So suppose we are given "I E r. Choose an-I ,. .. , ao E K with v(lii - ai) > max{-y,v(lii)} for i = 1, ... ,n - 1. Then v(ai) = v(lii). Then n-1 9 := X" + an_1X + ...+ ao E K[X] has a zero x E K. Using v(x) 2:: 0, we get 230 Appendix: Valued Fields Thus f(K) comes arbitrarily close to O. Q.E.D. Remark A.4.16: If (K, v) is algebraically maximal (A.3 .15), then (K, v) is relatively complete. Proof: (K,v) ~ (KB,VB) is an immediate, separable extension (A.4.13). So equality holds, by maximality. Q.E.D. Corollary A.4.17: If (K, v) is Henselian and char K = 0, then (K, v) is relatively complete. Proof: (A.3.19) and (A.4.16). Q.E.D. Remark A.4.18: The converse of (A.4.17) is in general not true. There are many complete valued fields (K,v) (of course, not ofrank 1, by (A.4.8), but still of characteristic 0) that are not Henselian." A.5 Dependence and Composition of Valuations Let 0 be a valuation ring of K, and let 0 1 be a subring of K. o C 0 =} {0 1 is a valuation ring of Kand 1 (A.5.0.1) - ml ~ m C 0 ~ 0 1 . Note that ml is also an ideal of OJ more precisely, it is a prime ideal of O. Claim A.5.1: 0 1 = Om! ' Proof: First, to see Om! ~ 0 1 , note that for a, b E 0 with b ¢ ml , we have 1 b- E 0 1 , whence alb E 0 1 • Second, equality will follow if the maximal ideal of Om! coincides with the maximal ideal of 0 1• The maximal ideal of 0 1 is mlj that of Om! is ml0mll which equals mI ' Q.E.D. Definition A.5.2: For valuation rings 0 and 0 1 in K such that 0 ~ 0 1 , we call 0 1 a coarsening of 0 in K. For the value groups r = K XlOX and r. = K X10f 4 Cf. Prestel, Ziegler [19781 . A.5 Dependence and Composition of Valuations 231 we have, using O x ~ Or: VI ~ K X ~ K XIO X ~ K XIor = r; '----..--" =r The subgroup r := Or[o? of r is convex; indeed, suppose a E or, X b E K , and 0 ~ v(b) ~ v(a); then b E 0 ~ 0 1 and alb E 0; then from a-I E 0 1 follows lib E 01-Le., se Or. Therefore Ti = r Ir, and the ordering on r 1 is induced by that of r (as one sees easily). Conversely, if r is a convex subgroup of r, then the mapping X VI : K ~ r 1 =rlr given by VI (a) := f(a) + r is a valuation on K with 0 1 = {a E K I VI (a) ~ O} = {a E K Iv(a) E r or v(a) > r}. In particular, 0 ~ 0 1 and r = Or [O", Hence this procedure furnishes a coarsening 0 1 of O. Next, we reverse this process. For this , we fix a valuation ring 0 1 of K, and consider a subring 0 ~ 0 1 , This time, however, we must require that 0 be a valuation ring, too. Then 0 := 0lm1 is a valuation ring of K:= OI/m1. Since mimI = m C 0/m1, we have Olm = 0/m1/m/m1 = ore: Therefore the valued fields (K,O) and (K,O) have the same residue field. The value group of (K,O) is here the equation marked with (!) holds since 0 = O x Um and 0 = (0) x Um implies (O) xUm=(OX)Umandthus (O)X =Ox. ~ Definition A.5.3: For valuation rings 0 and 0 1 of K such that 0 0 1 , we call 0 a refinement of 0 1 in K. In this case, 0 furnishes a valuation ring 0 on K = 0I/m1' Conversely, if 0' is a valuation ring of K, then 232 Appendix: Valued Fields o := { a E K I a + m1 E 0' } ~ 0 1 is a valuation ring of K that furnishes a refinement of 0 1 • Clearly 0 = 0'. Definition A.5.4: In this case 0 is also called the composition of 0 1 and O. Theorem A.5.5: Let (K,Ol ) be a valued field and 0 a valuation ring of K := 0t/m1. The composition (K,O) is Henselian if and only if both (K, 0 1 ) and (K,O) are Henselian. Proof: (=?): Suppose (K,O) is Henselian. Then (K,Ol ) is also Henselian, using m1 ~ m ~ 0 ~ 0 1 and (A.3.13)(6). To show that (K,O) is Henselian, let 7= X n + X n- 1+an_2x n- 2+ ...ao, with ai E m; we must show that 7 has a zero in K (again using (A.3.13)(6)) . The polynomial n n 1 2 f = X + X - + an_2x n- + ... + ao E O[X] has a zero x E 0 (yet again by (A.3.13)(6), since ai E m); therefore x E 0 is a zero of f. ({::): Let f = anxn + an_1xn-1 + an_2x n-2 + ... + ao E O[X], and suppose that f and hence also 7 has a simple zero z E O/m = ote. Since (K,0) is Henselian, 7 has a zero x in 0 with x+ m = z. In particular, x is a simple zero of 7 in K. Since (K,Od is Henselian, f has a zero a E 0 1 with a =x. From x E 0 we actually get a E O. Moreover, a + m = z . Q.E.D. Lemma A.5.6 : Let 0 ~ 0 1 be valuation rings of K. If 0 1 :/; K, then 0 and (~) 0 1 induce the same topology on K . In particular, the completions and ~) have the same underlying field. Proof: r-»- rtr = r, :/; {O}, since 0 1 :/; K. Write U-y(O) = {a E K Iv(a) > ')' } and U-y+r(O) = {a E K Iv(a) > ')' + r} = {a E K Iv(a) > <5 for all <5 with <5 == ')' mod r}. Then for')' > r, U-y+r(O) ~ U-y(O) and U2-y(O) ~ U-y+r(O), since v(a) > 2')' and v(a) ~ <5 for some <5 with <5 == ')' mod r would imply V 1 (a) ~ 2')' + T' and V1 (a) ~ ')' + r, contradicting ')' > r. Therefore the induced topologies are identical. Concerning the completions, observe: A.5 Dependence and Composition of Valuations 233 C'Yv) is cofinal in r ¢:} (-yv +1) is cofinal in F[I', Q.E.D . Definition A.5.7: Two valuation rings 0 1 and O2 of K are called dependent if their "product" 0 102 := 0d02] = 02[Od does not equal K. From the next theorem it follows that the dependence relation is an equiv alence relation on the set of nontrivial valuation rings of K. Theorem A.5.8: Two nontrivial valuation rings 0 1 and O2 of K are de pendent if and only if they induce the same topology on K. Proof: (=»: Let Oa be a common overring oj:. K of 0 1 and O2 • Then 0 1 and Oa induce the same topology (A.5.6), as do O2 and Oa. ( {=): Let m1 and m2 be the maximal ideals of 0 1 and O2 • If 0 1 and O2 induce the same topology on K, then there exists an a E 0 1 \ {O} with Set ma := rad(amd in the ring 0 1 , Note that ma is a prime ideal in 0 1 , since the ideals in 0 1 are linearly ordered (Exercise A.7.10)(i)) and rad(am1) is the intersection of all prime ideals lying over am1. Furthermore, ma ~ m1,m2 (since for all x E K, if x n E am1 ~ m2 for some n E N, then also x E m2). Setting Oa := (Odm3 ' we see that ma is the maximal ideal of Oa, and , by (A.2.7.1), Oa oj:. K, since ma oj:. {O}. Therefore 0 1 and O2 are dependent. Q.E.D. Consequence A.5.9: The dependence class [0] of a nontrivial valuation ring 0 in K is an upwardly directed set with respect to the partial order of inclusion. We distinguish between two cases. Case 1: There is a maximal valuation ring 0 1 oj:. Kover O. Claim A.5.l0: Then 0 1 has rank 1. Proof: r1 is Archimedean. For if there were a t5 E r1 for which Zt5 were not cofinal in r (and t5 > 0), then r := {'Y E r 1 111'1 < nt5 for some n E N} would be a proper convex subgroup of r 1; then X V2 : K -+ rl/r, a 1-+ V1 ( a) + r 234 Appendix: Valued Fields would define a valuation ring O2 2 0 1 with O2 =I 0 1 and O2 =I K. So r 1 can be order-embedded in (R, +). In this case, cK:0 = (~) and Vi is Henselian in R, by (A.5.6) and (AA.8). In particular, K is dense in the Henselization with respect to VI. Q.E.D. Case 2: There is no maximal valuation ring =I Kover O. Claim A.5.U: Then the maximal ideals m' of valuation rings 0' E [0) form a neighborhood system of o. Proof: Suppose we are given a positive 8 E r. We seek a valuation ring 0 1 20 such that 0' =I K and whose maximal ideal m' satisfies m' ~ U r ={'Y E r I I'YI < n8 for some n EN}. r is a convex subgroup of r, and defines therewith a valuation ring 0' 20 with m' ~ U Approximation Theorem A.5.12: Suppose 0 1 , ••• ,On are pairwise inde pendent valuation rings of K. Then for any aI, ... ,an E K and 'Yi E r(Oi) (1 :::; i :::; n), there exists an x E K with Vi(X - ai) > 'Yi , for all i E {I, ... , n}. Proof: For i = 1, ... , n, let m, denote the maximal ideal of Oi, and let X r i := riCO) = vi(K ). We choose 8i E r i so that 0 < 8i := 28i, 'Yi:::; 8i, and -8i :::; vi(ad, ... ,Vi(an ). Then we set u, = {x E K I s, < Vi(X)} and Ai = {x E K I -8i :::; Vi(X) }. M, and Ai are closed under addition and subtraction. (1) We may choose the 8: so that n M 1 n n(K \ Aj ) =10. j=2 Proof of (1): Induction on n. A.5 Dependence and Composition of Valuations 235 n = 2: If M 1 n (K \ A 2) = 0, then M 1 ~ A 2. Then if C2 E M 2 , it follows that C2M1 ~ C2A2 ~ m2' Then 0 1 and O2 would be dependent, by the proof of (¢:) of (A.5.8). n > 2: By the inductive hypothesis there exists r E M 1 n (K \ A 2 ). We choose the 8~, ... ,8~ large enough so that for j = 3, .. ., n, r E Aj. By the inductive hypothesis there further exists an s E M 1 n n (K\Aj ). 3~j~n If s rt A2 , we're done. If s E A2 , then s+r E M 1 n n (K\Aj ), 2~j~n proving (1). Analogously we find via "belated improvement" of the 8v that u, n n(K \ Aj ) :f. 0. # i An element from this intersection "approximates infinity" with respect to Vj for each j :f. i, and it approximates a with respect to Vi. (2) It now follows that (1 + M i ) n nu, :f. 0 #i (i.e., we can approximate 1 with respect to Vi , and a with respect to Vj for all j :f. i) j indeed, 1 x x E M, => -- = 1 - -- E 1 + M· and 1 + x 1 + x t, 1 x E K\Aj Vj(l + x) =Vi(X), whence -1- E M j • => +X (3) Then we choose di E (1 + M i ) n nu, #i and set finally Therewith follows Vi(X - ai) = vi(a1d1 + ...+ ai(di - 1) + ... + andn) > m~n{vi(aj) + 8d ~ 8i - 8: = 8: ~ Ii, 3 since di - 1 E M, and dj E M, for j :f. i. Q.E.D. 236 Appendix: Valued Fields A.6 Transcendental Extensions Theorem A.6.1: Suppose K is a field, T is an ordered subgroup of an ordered group I", v : K -1t ru{oo} is a valuation, and"( E I" , For f = L: ~=o aiXi E K[X], let (A.6.1.!) and for I.s E K[X) \ {a} let w(fjg) = w(f) - w(g). The above equations define a valuation w : K(X) -1t I" U {oo} on K(X) that extends v. Proof: First, (A.6.1.!) defines a map w : K[X] -t I" U {oo}, and for all f E K[X), w(f) = 00 if and only if f =a. Next, for f,g E K[X] \ {a}, let n = max{degf, degg}, and write f = L: ~=o aiX i and 9 = L:~=o i.x! (ai, bi E K). Then f + 9 = L:~=o(ai + bi)Xi, and v(ai + bi) + i"{ ~ min{v(ai), V(bi)} + i"{ = min{v(ai) + i"{ , V(bi) + i"{} 2: min{w(f), w(g)} , whence w(f + g) ~ min{w(f),w(g)}. (A .6.1.2) Next we show that for l.s E K[X] \ {a}, w(fg) = w(f) + w(g). This time write f = L:~=o aiX i and 9 = L:.i=o bjXj. Then v(aibj) + k"{ = v(ai) + i"{ + v(bj) + h 2: w(f) + w(g) =} v(aibj) ~ w(f) + w(g) - k"{ =} V(Ck) ~ w(f) + w(g) - k'Y =} V(Ck) + k"{ ~ w(f) + w(g). Therefore w(f) + w(g) ~ min (V(Ck) + k"{) = w(fg), O:::;k:::;n+m whence w(f) + w(g) ~ w(fg). (A.6.1.3) To show the opposite inequality, let i o = mini i Iv(ai) + i"{ = w(f) }, jo =min{j Iv(bj ) + h =w(g)}, ko = i o + jo· A.6 Transcendental Extensions 237 Then Cko = L aibj = ( L aibj) + aiobjo + L tub]. (A.6.1.4) i+j=ko i+j=ko i+j=ko i v(aibj) + ko'Y = v(ai) + h + v(bj) +if> w(J) + w(g), ~ > w(J) ------~ w(g) whence v(aibj) > w(J) + w(g) - ko'Y. As for the last summation in (A.6.1.4), we have i > io, whence i < i«. Then v(bj ) +if> w(g), by the definition of io. Then v(aibj) > w(J) + w(g) - ko'Y· But V(aiobjo) = w(J) + w(g) - ko'Y. Therefore Cko = aiobjo + L aibj i+j=ko i#io and v (~ aibj) > w(J) + w(g) - ko'Y = v(aiobjo)' t#to which imply V(Cko) = w(J) + w(g) - ko'Y. Therefore w(Jg) ~ V(Cko) + ko'Y = w(J) + w(g), which, together with (A.6.1.3), gives w(Jg) = w(J) + w(g), (A.6.1.5) as promised. Next we observe that w: K(X) -+ I" U {oo} is well defined, since gl g2 =} w(JI) + W(g2) = w(f2) + w(gI) (by (A.6.1.5)) =} w(JI) - w(gI) = w(f2) - W(g2)' It remains to extend (A.6.1.2) and (A.6.1.5) from the case of I,g E K[X]\ {O} to the case of arbitrary elements hI, h2 of K(X) \ {O}. For this, let 9 be a common denominator of hI and ha: hi = I.Is. where h,f2,g E K[X] \ {O}. Then 238 Appendix: Valued Fields fI + h ) W (hI + h2) = W ( 9 = w(fI + h) - w(g) ~ min{w(fI), w(h)} - w(g) =min{wUd - w(g), w(h) - w(g)} =min{w(hd, w(h2)}. Finally, w(hl h2) = w ( f~{2) = w(fIh) - w(l) =wUd - w(g) + w(h) - w(g) = w(hl) + w(h2), as required. Q.E.D. Corollary A.6.2: Suppose v : K ---* T u {oo} is a valuation of the field K , T is an ordered subgroup of an ordered group I", and"/ E T' has the property that 'In EZ (n,,/ E r => n = 0). (A.6.2.l) Then there is exactly one extension w : K(X) ---* I" U {oo} of v on K(X) with w(X) = "/. For this w , we have K(X) = K and w(K(X) X) = T (J) Z,,/. Proof: The existence of w follows from (A.6.l). To prove uniqueness, let w be any such extension. Consider an f E K[X], say, f = ao + alX + ... + anxn, with ai E K. Then for each i $ n, We claim that (A .6.2.2) Indeed, otherwise ai ::J O::J aj and v(ai) + h = v(aj) + rt. whence (i - in = v(aj) - v(ai) E r, whence (by (A.6.2.l)) i - j = 0, i.e., i = j. Now (A.6.2.2) and (A.1.5)(3) yield which implies that w is uniquely determined on K[X], and hence on K(X). It is now clear that w(K(X) X) = r(J)z,,/. It remains to show that K(X) = K. First we show that every f E K[X] \ {O} is of the form f = axm(l + u), where a E K X, mEN, u E K(X), and w(u) > O. For this, write f = E~=o aiX i, with ai E K. There is exactly one io such that A.6 Transcendental Extensions 239 wU) = v(aio) + h = w(aioXiO), by (A.6.2.2) and (A.1.5)(3). Therefore Observe that aiXi ).. w ( -X' = w(ai X t) - w(aioX tO) > 0 ato ' to for i :j:. i o, whence w(u) > O. Second, we consider any h E K(X) \ {O}, and write h = fig, with t.s E K[X] \ {O}. Write f = aXm(l + u) and 9 = bxn(l + u'), with a, b E KX, m,n E N, and w(u),w(u') > O. Then h = £ = cXm-n 1 + u = cxm-n(l + u - U') = cXr(l + u"), 9 1 +u' 1 +u' X where c = alb E K , r = m - nEZ, and " u - u' u =--. l+u' Since w(u') > 0, w(l + u') = 0; therefore w(u") > O. Finally, to show K(X) = K, we show that for any h EO;:', Ii E K. We have 0= w(h) = w(cXr (1 + u"» = v(c) + r'Y, whence r"f = -v(c) E F; then r = 0, by (A.6.2.1), and then v(c) = O. Therefore h = c(l + u"), whence Ii = c(1 + u") = c (since u" = 0); i.e., Ii E K. Q.E.D. Corollary A.6.3: Suppose v : K -» T U {oo} is a valuation on K. Then there is exactly one extension w of v to K(X) such that w(X) = 0 and X is transcendental over K . For this w, we have K(X) = K(X) and w(K(X)X) = r. Proof: For the uniqueness, let f = L:~=o aiXi E K[X] \ {O}. Pick k ~ n such that Then n f = ak LbiXi, (A.6.3.1) i=O ~ =:g 240 Appendix: Valued Fields Then w(g) 2': 0, since w(X) = O. Moreover, n g= LbiX #0, i=O since bk = 1 and X is transcendental over K. Therefore 9 E Q~ , i.e., w(g) = 0, whence w(J) = v(ak), i.e., (A.6.3 .2) For the existence, define w(J) by (A.6.3.2) for I E K [X] \ {O}, accord ing to (A.6.1). Then w(X) = O. To see that X is transcendental, suppose n --=i - Ei=l ai X = 0, for some ai E Q v • Then whence v( ai ) > 0 for each i; i.e., each ai = O. Next , W(K(X)X) =r is clear. The last property to show is that K(X) = K(X). For this, let h E Q~ , and write h = II!12 , with II,12 E K[X] \ {O}. As in (A.6.3.1), write I i = cigi (i = 1,2), where c, E K X, gi = E ?=oaijXj , and for all i , v(aij) 2': 0, while for some j (depending on i), aij = 1. Then each gi E Q~ , as before. Also, gl X h=c-, where c = Cl E K • g2 C2 Therefore g2h = Cgl . Also, c E Q~, since h E Q~. Then from g2h = Cgl follows h = Cgl g2-1 E K(X) . Q.E.D . Definition A.6.4: For an Abelian group G, we define rr( G) := sup] n E N I 3al , ... , an E G, linearly independent over Z } to be the rational rank of G. Example A.6.5: (a) If G is finite, then rr(G) = O. (b) rr(Z ) = 1, rr (Q) = 1, rr( Zn) = n . (c) rr( IR) = 00, since, e.g., 1,11",11"2, •• • are Z-linearly independent, by the transcendence of 11" . Theorem A.6.6: Suppose K'/ K is a field ext ension, v : K -» T u {oo} is a valuation on K, and v' : K' -» I" u {oo} is an extension 01 v to K'. Let Xl, .••, X s E QVI be such that Xl, •.• ,X s E K' are algebraically independent A.6 Transcendental Extensions 241 over K . Further let Yl, .. ., Yr E K'x be such that V'(Yl), . ..,v'(Yr) E I"/ r are Z -linearly independent. Then Xl, .. ., Xs, Yl , ... , Yr are algebraically inde pendent over K. In particular, tr.deg.(K'/K) + rr(r'/ r) s tr.deg.(K'/ K). Proof: Since Xl, ..., Xs are algebraically independent over K, Xl is transcen dental over K. Therefore Xl is transcendental over K, for if K (xd/ K were algebraic, then K(Xl)/K would be, too (by (A.2.4)). So Xl EO:, (otherwise Xl = 0). By (A.6.3), V'IK(xtl is the uniquely determined extension w of v to K(xd with w(xd = 0 and Xl transcendental over K. Also by (A.6.3) we conclude that K(xd = K(xd and v'(K(xd X) = r. Similarly, X2 is transcendental over K(xd = K(Xl), whence X2 is tran scendental over K(xd, whence X2 E 0;'. The uniqueness stated in (A.6.3) gives V'(K(Xl,X2)X) = rand K(Xl,X2) = K(xd(X2) = K(Xl,X2). Iteration of the above leads to in which each extension is transcendental. Therefore Xl, .. ., X s are alge braically independent over K. Furthermore, v'(K(Xl, ... , X s)X) = T and K(Xl," "Xs) =K(Xl" "'Xs), Next, for n EZ\ {O} we have nv'(Yl) ~ r, since V'(Yl), .. . ,v'(Yr) are Z-linearly independent. Therefore VI is transcendental over K(Xl, "" xs) (otherwise V'(K(Xl,,,,,Xs,Vl)X)/r would be a torsion group by (A.2.4)). The uniqueness statement in (A.6.2) gives K(Xl,' .. , XS , yd = K(Xl,' .. , xs) and V'(K(Xl,'" ,Xs,Vl)X) = T + Zv'(vd. Similarly, 'hV'(Y2) E r + ZV'(Yl) only for n = 0, whence Y2 is tran scendental over K(Xl, ' .., XS , Yl)' The uniqueness statement in (A.6.2) gives K(Xl, ' " ,Xs,Yl ,Y2) = K(Xl, '" ,xs,Yd = K(Xl, '" ,xs) and Iteration of the above leads to and each extension is transcendental. Therefore VI , . . . , Vr are algebraically in dependent over K'(Xl, . .. , xs), whence Xl, .· ., Xs, VI, ... , Vr are algebraically independent over E. Q.E.D. Theorem A.6.7: Suppose K'/ K is a field extension, 0 is a valuation ring of K, and 0 1 C. .. C On are extensions of 0 to K' (where C denotes proper inclusion) . Then tr.deg.(K'/ K) ;:::: n - 1. 242 Appendix: Valued Fields Proof: To 0 belongs a valuation v : K --» T u {oo}; and to 0 1 belongs a valuation v' : K' ~ T' u {oo} with r ~ I", Choose Since u. f/- Oi-1, y:;1 E Oi-1 C Oi, whence Yi E 0: C 0:+ 1 C ... c 0;: . By (A.6.6), it will suffice to prove that V'(Y2), . .. ,v'(Yn) E I"/ r are Z linearly independent. For the latter, suppose, on the contrary, that for some k2, ... , k« E Z, not all o. Then k2v'(Y2) + ... + knv'(Yn) = veal, for X some a E K • Let m := max{ i 12 ~ i ~ n, ki "# O}. Then whence k2 yk", b .- Y2 .. . m E O x - O x .- a v' - 1· Then a = b-1y~2 ... Y~"' E O~ n K = O x (since Om n K = 0). Moreover, k b -k2 -k",_l E Ox h Ox c 0 (. Ym"' = aY2 ... Ym-1 m-1' W ence Ym E m-1 _ m-1 SInce »; "# 0). Contradiction. Q.E.D. A.7 Exercises A. 7.1 Suppose L / K is a finite extension of fields, v : K ~ r u {oo} is a valuation on K that is discrete and of rank 1 (i.e., r ~ Z), and w : L ~ I" U{oo} 2 T is an extension of v to L. Show that w is also discrete and of rank 1. A.7.2 Let K be a field, and R a subring. Show that the intersection of all valuation rings of K containing R is the set of all elements of K that are integral over R. (An element of K is called integral over R if it is a zero of a monic polynomial in R[X], where X is a single indeterminate.) (Hint: If z E K is not integral over R, then the ideal generated by l/x in R[l/x] is proper.) A.7.3 Let 0 be a Henselian valuation ring of K with residue field K. Sup pose char K = 0, and let p : 0 -t K denote the residue map x f-+ x. Moreover, let L be a common subfield of K and K (e.g., L = (11) such that plL = idj.. Show that there is an embedding a : K -t () with po a = idK and aiL = idL. (This result is used in the proof of (2.4.2).) A.7 Exercises 243 (Hint: Choose a maximal subfield L 1 2 L of K together with an embedding al : L 1 -t K such that po al = idL1 and allL = idL. Now prove L 1 = K.) A .7.4 Let v : K -*r U {(X)} be a valuation of K with valuation ring O. Assume that a E Aut K fixes 0 as a set, i.e., a( 0) = O. (i) Show that there exists a unique order-isomorphism p : T -t T such that p(v(x)) = v(a(x)) for all x E K X. (ii) Consider the field K of all quotients from the ring of polynomials Q[.. . ,X-1 ,XO,X1 •••Jin X i (i E Z) together with the automor phism a sending Xi to Xi+!. Define a valuation v : K -t ZZU{ 00} by assigning to each polynomial f the negative of the minimal exponent of monomials in f, where the exponents are ordered lexicographically from left to right. Show that in this case p =I id. (iii) Show that p = id if, e.g., a has finite order (i.e., an = id for some n E N) or if there is a subfield F of K such that alF = id and v(F) = v(K). (iv) Find other sufficient conditions for p = id. A.7.5 Let v : K -* r u {(X)} be a Henselian valuation on K such that the corresponding residue field K is not of characteristic 2. Show that for X each a E K , the following conditions are equivalent: 2 (i) a E K j 2 (ii) v(a) E r, and for each bE K with v(a) = 2v(b), a/b2 E K j 2. (iii) there is abE K with v(a) = 2v(b) such that a/b2 E K Show, in addition, that if (v(ai) +2r)iEI is an 1F2-basis of r/2r (for X)2)jEJ some index set I and for some ai E K X), and if (bj(K is an X X) 1F2 -basis of K / (K 2 (for some index set J and for some bj E O X), then «ai(KX)2)iEI, (bj(K X)JEJ) X X is an lF2-basis of K /(K )2. In particular, there is an (in general noncanonical) group isomorphism In Exercises A.7.6-8 below, let v : K"""* T U {(X)} be a nontrivial valuation on K , and let /'i, be the cofinality of r (recall (A.4.9)). Readers not familiar with transfinite ordinals may restrict to the case /'i, = No (so that (Yi)i<1< = (Yi)iE N). A.7.6 Let (ai)i<1< be a Cauchy sequence in K (with respect to Vj recall (AA.I0)). Show that exactly one of the following conditions holds: 244 Appendix: Valued Fields (i) there is (exactly) one"( E r such that 3i < K, Vj ~ i, v(aj) = ,,(i or (ii) V"( E r, 3i < K, Vj ~ i, v(aj) > "(. Show, in addition, that if (ai)i A.7.S Let Land L' be completions of the valued field K. Show that there is exactly one isomorphism ¢ : L -+ L' of valued fields with ¢IK = idK. A.7.9 Let 0 be a valuation ring of the field K. Show that the prime ideals I:J of 0 correspond bijectively to the subrings 0 1 ~ K containing 0 , by means of the mappings where m1 denotes the maximal ideal of 0 1 • For every prime ideal p of 0, we have p = pOp. A.7.10 Let v : K -»- r u {(X)} be a valuation on K. Let t; := {"( I"( ~ O}, r:r := Ts: u {oo}, and 0 = v- 1 (r:r ), the valuation ring belonging to v. (i) We call a subset G ~ r:r an upper subset of r:r if for all "( E G and "(' E r:r, "( ~ "(' =} "(' E G. Show that the ideals a of 0 correspond bijectively to the nonempty upper subsets G of r:r, by means of the mappings 1 a I-t v(a) and G I-t v- (G). In particular, the set of ideals of 0 is totally ordered by inclusion. (ii) Show that the prime ideals of 0 correspond bijectively to the convex subgroups Ll of r, by means of the mappings A.8 Bibliographical Comments 245 (iii) Show that the subrings 0 1 ~ K containing 0 correspond bijec tively to the convex subgroups Ll of r, by means of the mappings A.7.11 Let K be a field, and X an indeterminate. Determine all nontrivial valuations on K(X) that are trivial on K. Moreover, show that the only possible value group is Z, and the only possible residue fields are either K itself, or K[X]/(P) with p irreducible in K[X]. (Hint: Let v : K(X) -+t r U {oo} be a nontrivial valuation. Distinguish the cases v(X) < 0 and v(X) ~ O. In the first case, investigate the behavior of v on K[X], using v(J) < v(g) => v(J + g) = v(J). In the second case, first show that if f, 9 E K[X] are irreducible and v(J) f 0 f v(g), then f and 9 are associates.) A.7.12 Construct (a) on Q(X) a valuation with value group Z and residue field lFs (t, v'2) (t transcendental over lFs), and (b) on IR(X, Y) a valuation with residue field C and value group Z xZ, where the ordering on Z x Z is prescribed. A.7.13 Give a direct proof of Theorem A.6.7 in case 0 = K. (Hint: Apply the following statement, which can be proved by induction on n E N: Let A be an algebra over the field K with prime ideals Po ~ 1'1 ~ ... ~ Pn, and let ai E Pi \ Pi-1 for 1 ~ i ~ n . Then a1, ... ,an are algebraically independent over K .) A.S Bibliographical Comments The reader who wants to learn more about valuation theory is referred to the books of Bourbaki [1962], Endler [1972], Kuhlmann [2001 (preprint)], Ribenboim [1968, 1999], and Schilling [1950]. More on topological properties of valued fields can be found in Prestel, Ziegler [1978] . In particular, the notion of a "topological Henselian field" is introduced there. References F. Acquistapace, C. Andradas, and F. 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Siegel [1921] Darstellung total positiver Zahlen durch Quadrate, Math. Z. 11, 1921, 246-75; Ges. Abh. 1, (1966), 47-76 . G. Stengle [1974] A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Math . Ann. 207 (1974), 87-97 . [1996] Complexity estimates for the Schmiidgen Positivstellensatz, J. Com plexity 12 (1996), 167-74. M.H. Stone [1940] A general theory of spectra. I, Proc, Nat . Acad. Sci. U.S.A . 26(4) (1940), 280-3. A. Tarski [1948] A Decision Method for Elementary Algebra and Geometry, Rand Corp ., 1948; UC Press, Berkeley, 1951; announced in Ann. Soc. Pol. Math., 9 (1930; published 1931) 206-7; and in Fund . math., 17 (1931) 210-39 . Page proofs of The completeness of elementary algebra and geometry were prepared in 1940 for Actualites Sci . Indust. , but not published until 1967, by Inst. B. Pascal. E. Witt [1937] Theorie der quadratischen Formen in beliebigen Korpern, J. reine angew. Math . 176 (1937), 31-44. Coli. Papers, ed. by I. Kersten, Springer, 1998. T. Wormann [1998] Strikt positive Polynome in der semialgebraischen Geometri e, Disserta tion , Univ. Dortmund, 1998. Glossary of Notations < used to compare subsets, 8n B(Ch,n, s,d, N) (bound on degree), :5 determined by a (pre)positive cone, 1 193 :5P, 10 B(n, s, d) (bound on degree), 185, 187, +,-,' used for subsets of a ring, IOn 189 ..1 (sum of quadratic forms), 56 B(n, s, d, N) (bound on degree), 195, ® (product of quadratic forms), 59 199,200 ~ , ~K (equivalent quadratic forms), iC (complex numbers), 60 54 C(R(N), R), 93 11·11 (Euclidean norm), 47, 104, 117, C(X(K), Z) , 66 156, 191-195, 199, 200 C(X, R), 1, 4, 5, 116 11 ·11 (sup of Euclidean norm), 190-195 C(X,RZ ), 116 II · II (sup-norm on C(X, R)), 153 C(XA'tX,R), 127, 169 Z ~ (for ultraproducts), 37 C(XA'tX,R ), 127, 170 Z ~, ~K (similar quadratic forms), 59 C(X~a.x, R ), 121 -+ , B vs. =?, {::}, 44n card (cardinality), 95 /\ , V, ., (and, or, not), 32n charp (characteristic of p), 61 OW(K) , l w (K) (identities of W(K)), 60 D(p) (value-set of p), 69 17th problem, see Hilbert, D. Dn.d ~ R(N), 91 2-group, see group ..1 (difference), 176 A Z (set of squares), 1, IOn 8 or 8(Xl ,..., X n ), 32 A = A/p, 83, 101 8(a) ("8 holds at a"), 32 A(n) vs. An, IOn o(R) (truth-set of 0 over R), 32 Aut, 22 0[8] or 0[8](Xl" .. ,Xn ), 38 An := s:'A, 109 det (determinant), 54 Nl-saturation, see saturation e(Oz/OI) (ramification index), 207 N", -saturated real closed field, see ef :5 n, 209 1]",-field Fo (filter of cofinite sets), 37 a = (al, , a n) E (R" )(n), 102 lF2 (field with two elements), 60 lFl' (field with p elements), 225 a : R[X l, , X n ] -+ R" , 102 a" : R[X l, ,Xn]-+R", 102 7 (residue of I), 27 a ~ f3 (specialization), 103 7 (similarity class of I), 60 ap : A~ A/suppP, 83, 86, 101 J: Sper A -+ R" , 2 as : A -» A/supp S, 115, 124, 164, f(OZ/Ol) (residue degree) , 207 169, 197 !' (formal derivative), 17 ap := (a) ® p, 68n (1,1') (gcd) ,17 a: X~a.x -+ R, 121 f..1g (sum of quadratic forms), 56 (al , ,an ), 54,172 f ® 9 (product of quadratic forms), 59 «al, ,an)), 69 t.: E Z[CjX], 91 (al , ,an)" , 141, 172 f(P) E 2Z, 78 «al, ,an))Zm , 176 f(P) E A/suppP, 2, 86, 115 256 Glossary of Notations I(S) E AI supp S, 115 ti] (multiple of a quadratic form), 56 ITG ~ A, 86 Nil A (nilradical of A), 65 Gal(LIK), 211 V Imo (residue field), 26, 203 111-field, 48 V(B, ::;), V(R, ::;), V(Z, ::;), 12, 26, 111-ordering, 48 203,221 l1o-field, 94-96 V ~ JR' , V(P), V(S), V(T) - embedding into, see embedding (convex hulls of Z), see also valua H(a) (Harrison basic subset), 72, 78 tion, ring, canonical, 102, 120, 141, HomT(A,JR), 121 166 ,fl (radical of I), 65, 77 V v (valuation ring of v), 204 - in addition, see V'T and rrad I V v p = Z(p) (valuation ring of vp ), 203, 204 /(P) , /A(P), 1m) , /(/1, . .. , ~ /A(/I, .. . ,1m) ~ A, 86 ITG A, 86 /(K) (fundamental ideal of W(K» , 60 IT.Es KI.]IF, 37 Irr(o; K) (irreducible polynomial), 13 P (Ka, va) (relative separable closure), - kernel 228 -- oftjl :B-tJR, 167 K S IF (ultrapower of (K, ::;» ,38 - - of p : A -t P', 45 - prime ideal K := Vim = Vlmo (residue field), 26, 203 -- of A, 84, 142 K((G» (formal power series field), 25 - - of A containing a, 109 -- of A disjoint from S, 108 (I(,0 := (K,v) (completion), 225, 228 - - of V, 189, 190, 192, 196, 197, 210, (K, P) (real closure), 21 244 (Kz, V' n Kz) (Henselization),213 - - of R, 205, 210 Kz (decomposition field), 213 -- of W(K), 61 K+ (strictly positive elements), 89, 97 - real prime ideal K 2 (set of squares), 10 - - of A, 141, 142, 144, 148, 150, 151, K X (nonzero elements), 12 172,173,176 K· (separable closure) , 213 - - of A containing I, 88 K(n) vs. x» vs. K[n], lOn, 36n - support tc, (residue field), 27, 48 - - of a module of level 2m, 162, 164 k((X - e» (formal Laurent series), 226 -- of a (pre)ordering, 81-83 , 101, 106 LIK (field extension), 12 - - of a semiordering of level 2m, 169, [L : K] (degree of Lover K), 13 172 L : A -t JR, 152 -- of a (T-)semiordering, 115, 141 PP := ker(O'p 00'), 61 L : JR[X1, .. ., X n ] -t JR, 152, 154, 158 p(A) (Pythagoras number of A), 179 L : JR[XJl -t JR, 155 n - p(R(X1, ... ,X 2 , 69 PM(A) -t JR, 153 n »::; L: - p(R[XJ) 2, 180 leU) (leading coefficient of f), 134 = - p(R[X1, .. . ,X )) = 00,180 lim an , 24, 224 n n-+oo - P2m(K) (2mth Pythagoras number), l(O') (length of 0'), 179 178 - l(O') ::; (n~d) , 182 - P4(R(Xl» ::; 6,178 mo := m ~ V (maximal ideal of V), p+ := P \ suppP, 115 26,203 P = Pip, 83, 101 uo« ,... ,h.), M(h), 5, 113, 119, 142 Po ~ JR[X1 ,•••, X n ], 102 2m(h M 1 ,••• ,h. ), 172 p$ (positive cone of ::;), 10 M] (matrix of a quadratic form), 53 Pp (positive cone of p E Spec W(K», mdeg (multi-degree), 134 64 N (natural numbers), 2n P : x~ax -t HomT(A, JR), 121 Ns(x) (number of sign-changes at x), PM : A -t C( x~ax, JR) , 5, 127, 152, 169 18 PT : A -t C(x~ax,JR) , 4, 121 Glossary of Notations 257 1> or 1>(X1,...,X n), 34 sgnp : W(K) -t Z, 63 1>(R) (truth-set of 1> over R), 34 sgnj, p (signature of p), 62, 63 1>(a) ("1> holds at a"), 34 Spec A (Zariski spectrum of A), 61 1>['] or 1>['](X1,.. . , X n), 38 Sper A (real spectrum of A), 2, 84 1>1 (symmetric bilinear form of f), 54 Sper R[X]II ~ SperR[X], 107 Q (rationals), 8 Sperm ax A, 3 Q (algebraic closure of Q), 42 - compactness of, 107 - found by Krivine, 109 Qp := tJ;,), 225 (0, suppT (T any subset of A) , 81 Q(X ) 1 - in particular, see module, - some orderings on, 9 (pre)ordering, preprime, or Q+ (positive rationals), 9 semiordering Qp,120 Quot(A), 83, 109 'iff (set of mth roots), 175 Quot(P),83 - in addition, see ..;Iand rrad I ~ ryt(p), ryt1(p) , ryt~(p), 142 T(H), T(h1 , •• •, h.), T(h) A, 3, 77, R (reals), 1 86, 118, 145 R' , 2, 36, 94 tr.deg. (transcendence degree), 96 - "compactness property" r» : R[X1, . . . , Xn]-t R' , 101 (N1-saturation), 40, 94 U(a) ~ Semi-Sper A, 119 R+ (positive reals), 8 U(a) ~ Sper A, 85 R(X1) U.,(a) ~ K, 223 - orderings of, 8-9 U(f) ~ Sper R[X1, . .. ,Xn], 104 R', 40, 42, 44, 48 UR(f) ~ R(n), 31 - uncountability of, 41 (u,v) (open interval), 17 R2 (set of squares), IOn [u,v] (closed interval), 17 R S IF (ultrapower of R), 40 V(H) ~ Sper A, 88 R(n) vs. R", IOn VR(/l, 00 " 1m) ~ tc», 90 Ro (real algebraic numbers), 42 VR(I) ~ R(n), 106, 157 P 1l := 0p/p, 190 VR.(I) ~ (R·)(n), 106 rr(G) (rational rank of G), 240 Voo := k(X) -t ZU {oo}, 204 rrad I (real radical of 1), 88 va := VIKo, 228 - in addition, see ..;I and 'iff vp : Q -t ZU {oo}, 204, 224 2 L:~ := L:K+. A , 90 W(K) (Witt ring of K), 53, 60 2m 2m Wred(K) (reduced Witt ring), 66, 79 L: := L:A , 161, 170 L: A 2 (set of sums of squares of A), 1 Wt(K) (torsion subgroup of W(K», E K 2 ~set of sums of squares of K), 11 66 W(H) ~ Sper A, 88 E R[X] m (sums of 2mth powers), 5 ~ S = SIp, 115, 164 WR(h 1, oo .,h.), WR(h) R(n), 3, 77, ~ 90, 142 S SperR[X1, . .. ,Xn], 104 - bounded (= compact if R is R), 4, ~ S· (R' )(n), 104 5, 113, 118, 123, 124, 129, 133, 137, S+ (righthand half of a cut), 97 139, 144-146, 148, 150-155, 157-159, S+ := S \ supp S, 115, 164 170, 173, 174, 176, 187, 189, 192, 194, S-1 := ((S+)-1, (3.t:) - 1), 98 199, 248, 250, 253, 254 S-1 A (rinyof fractions of A), 108 X(K) (space of orderings), 27, 65, 78 S-1p ~ S- A, 108 XM := XM(A) S-2 P ~ S-1 A , 108 (M an arbitrary subset of A) , 115 Semi-Sperj- A, 114 Xl:tx, 4, 5, 115 T sgnp 1ri (sign of 1ri), 28 x (transpose of x), 54 sgn : W(K) -t C(X(K), Z), 66 }J(K) (space of semiorderings), 135 sgnp : X(K) -t Z }JM := }JM(A) (total signature of P), 65 (M an arbitrary subset of A), 115 258 Glossary of Notations 2m 1117 (A) , 1l (A) , 163, 169 Z (p)= CJv p , 203, 204 1l~ax, 115 Z (integers), 8 Z(CJ') (decomposition group), 213 Index absolute value, 16n in addition, see abstract B(n, s, d), B(Ch,n, s, d, N) , and - Positivstellensatz, see B(n,s,d,N) Positivstellensatz, abstract - on roots, 19 - real Nullstellensatz, see bounded, see Nullstellensatz, real, abstract WR(h1, . . . ,h.), WR(h) ~ R(n) Acquistapace, F., 159 Bourbaki, N., 245 affine, see variety Brocker, L., 48, 49, 79, 110, 137, 158 algebra Brocker-Prestel Local-Global Principle, - real, V, 1 158, 177 algebraic, see valuation, extension of a Brocker-Scheiderer Theorem, 49, 79 - geometry, real, 3, 110 Broglia, F., 159 - set, 32 Browder, F., 250 algebraically maximal, see valued field Brumfiel, G., 48 Andradas, C., 48, 49, 110, 159 anisotropic, see quadratic form Cancellation Theorem, see Witt, E. Approximation Theorem, see valuation, canonical, see ordering, preordering, ring topology, or valuation ring Archimedean ring, or Archimedean .. ., Cassier, G., 137 see module, ordering, ordered field, Cauchy preordering, or preprime - complete, see complete, Cauchy Artin, E., V, VI, 2, 3, 14,21, 29, 31, 35, - sequence, 24, 224, 228 36, 49, 51, 75, 79, 81, 90, 109 Cemikov, S.N., 137, 188 Artin-Lang Theorem, 49 Chang, C.C., 40, 49, 110 Aut, 22 characteristic, see prime ideal or automorphism group, 22 ordered field Characterization Theorem I, 142, 158, Backmeister, T ., 111 172 Baer, R., 29 Characterization Theorem II, 144, 158, Baer-Krull correspondence, 29 173 basic closed, basic open, see Chevalley's Theorem, see semialgebraic, set valuation, ring basis Chinese Remainder Theorem, 211 - orthogonal, 54 Choi, M.D., 178, 201 Becker, E., 79, 137, 177, 178 Christensen, J.P.R., 155, 159 Berg, C., 155, 159 closure Berr , R., 178 - real , see real closure Bochnak, J ., 48, 49, 110 - under specialization, 105 Borel measure, see measure coarsening, see valuation, ring bound cofinal, 26, 228 - on degree, 179 cofinality, 228, 243 -- existence of, 183-189 cofinite sets, filter of, 37 260 Index compactness - cut, see cut, Dedekind - of Sperm ax A, 107 definite quadratic form, see - property of JR" , 36, 40 quadratic form - theorem of model theory, 49 definition - in addition, see - prenex, 34-36 WR(h1,o . . ,h.), WR(h) ~ R(n) - quantifier-free, 32 complete - semialgebraic, 32, 34, 35 - Cauchy, 24 degree, see bound on - cut, see complete, Dedekind Delzell, C.N., 49, 51, 110, 111, 201 - Dedekind, 9, 10, 24 dense subfield , 8 - relatively dependence class, see valuation, ring - - valued field, see valued field dependent, see valuation, ring - valued field, see valued field diagonal, see form of degree 2m, completion, see valued field, 225 and quadratic form composition, see valuation, ring difference of a polynomial, 176 cone, see positive cone or prepositive dimension, see quadratic form cone discrete, see valuation Conjugation Theorem, see divisible , 27 valuation, extension of a van den Dries, L., 49 constructible Dubois, D.W., 4, 109, 110, 137 - set, 85 Efroymson, Go, 49 - topology, see topology Elimination of Quantifiers, see Tarski continuous solution to Hilbert's 17th A. ' problem, see Elman, n., 79 Hilbert, D., 17th problem embedding convergence, 228 - into 110-fields, 81, 95, 96, 110 convex - into JR, 9, 28 - hull, 12 - of JR(n) into Sper JR[X , ••• ,X ], 102 - subgroup, 24 1 n - subring, 12 - of function fields into JR" , 81, 96 Endler, 0., 245 Cornelsen, So , 111 equivalent, see quadratic form Coste, M., 48, 49, 109, 110 errata, VI Coste-Roy, Mo-F., see Roy, M.-F. Euler, L., 29 countably represented cut, 97 exponential function, 47 critical, 194 extension, see cut, 97 field, ordering, or valuation ring - complete, see complete, Dedekind - countably represented, 97 factorization of polynomials, 16 - Dedekind, 9 field - group, 98 -111-,48 - positive, 97 - 110 -, 94-96 - valuation, 99 -- embedding into , see embedding - algebraically closed, 15, 16 Dai, Z.D., 201 - embedding of a, 17 Daykin, DoE., 201 - extension of a decomposition -- algebraic, 16 - field, see - - Galois, 15 valuation, extension of a -- odd-degree, 13-14 - group, see -- quadratic, 13 valuation, extension of a -- transcendental, 14 - theorem, see Witt, E. - maximal ordered, see ordered field Dedekind - of characteristic not 2, 53 - complete, see complete, Dedekind - of fractions, 42, 74, 109 Index 261 - ordered, see ordered field -- rational rank of an , 240 - Pythagorean, 78 - in addition, see - real , 12, 29 ordered group; or ordering, group - real closed, 14, 29 Grundlagen der Geometrie, see Hilbert - SAP, see SAP D. ' - uniquely orderable, 14 field ordering, see ordering, field Hahn, H., 28 filter , 37 Handelman, D., 137 - of cofinite sets, 37 Harrison, D.K., 79, 249 finitely ramified, see valued field - basic subset, 72, 78 Finiteness Theorem, 44-48, 106, 249 - topology, 78 form of degree 2m ("diagonal"), 172 Hausdorff, F., 110 - 2m-isotropic, 172 Hensel's Lemma, 218 - Pfister, 176 Henselian, see valued field - regular, 172 Henselization (or Henselian closure), - regular part of a, 172 see valued field - weakly 2m-isotropic, 172 Hilbert, D., V, 2, 28, 29n, 49-51 - in addition, see quadratic form - 17th problem, V, VI, 2, 31, 36, 49, form of degree d, 49 51, 74, 75, 81 formal -- or variation in solutions of, 111 - deduction, 185 -- continuous solution, 31, 81, 91-94, - derivative, 17 110, 111 - Laurent series, 226 -- generalization, 3, 53, 74-77, 90 - power series, 25, 28 -- generalization (abstract version) , formally real, 29 88 fraction - Grundlagen der Geometrie, 28, 50, - field, see field of fractions 51n - ring, see ring of fractions Hodges, W ., 49 Fuchs, L., 25 Holder, 0 ., 28 function Hoffmann, D., 80 - semialgebraic, see hyperbolic, see quadratic form semialgebraic function ideal function field, real - real, 84 - in n variables, 80 immediate extension, see -- embedding into JR" , 96 valuation, extension of a - in one variable, 72 indefinite quadratic form, see -- SAP, 72 quadratic form -- Witt's Local-Global Principle, independent, see valuation ring 73-74 integral, 242 functional integral domain - analysis, 4, 136, 137 - ordered, see - linear, 152, 154, 155, 158, 159 ordered integral domain fundamental ideal, see Witt ring International Congress,V interval, 17 Galois group, 15 -, see topology Gauss extension, see valuation irreducible polynomial, 13 general polynomial, 91 isotropic, 2m-isotropic, see Godel's Completeness Theorem, 185 form of degree 2m, and quadratic Gonzalez-Vega, L., 110, 111 form Greenberg, M., 68 group, 24 Jacobi, Thomas, 124, 127, 137, 153, - 2-group, 15 158, 159, 170, 177 - Abelian Jacobson, N., 68 262 Index Kadison, R.V., 4, 109, 247 model Kadison-Dubois - complete, 49 Representation Theorem, see - theory, 110, 201 Representation Theorem module Keisler , H.J. , 40, 49, 110 - T- (T a preordering), 113 Knebusch, M., 29, 79 -- generated by al, ... , am , 113 Krivine, J .-L., 3, 109, 110, 137, 159 -- maximal, 114 - maximal real spectrum, 109 -- support of a, 114 - Positivstellensatz, 109 - T- (T of level 2m), 161 - real Nullstellensatz, 109 -- support of a, 161 Krull, W. , 29, 249 - of level 2m, 161 - dimension, 143 -- Archimedean, 164 - valuation, see valuation - quadratic, 4, 113, 115 Kuhlmann, F.-V., 245 Archimedean, 5, 116, 129, 144 Kuhlmann, S., 159 - - of lR[Xl, .. . , x n ] := lR[Xl , . . . , X n ]/ I, 117 Lagrange, J.L., 50 moment problem, 152-157, 159 Lam, T.Y., 29, 79, 80, 159, 178, 201 monoid Landau, E., 29, 29n, 49n - multiplicative, of a ring , 86 Lang, S., 49, 68, 247 multi-degree, 134 Laurent series multiplicative monoid , see monoid - formal, 226 multiplicative set, 108 leading coefficient, 134 nilradical of a ring, 65 Leicht , J., 79 non-critical, 194 length of sum of squares, 179 nonstandard, 118 level 2m, see module, (pre)ordering, or normal subgroup, 24 semiordering n-types, space of, ~ SperlR[Xl, ... , X ], linear n 110 - functional, 154 Nullstellensatz - ordering, see ordering - real, 81, 90, 110, 250, 254 - representation, 4 -- abstract, 88, 109 linear representation, 139, 152 - - due to Krivine, 109 Local-Global Principle, see Brocker- number field, 29n Prestel, Pfister, or Witt Local-Global Principle, homogenous, Oberwolfach, VI 146 order-embedded, 9 local ring, see ring order-embedding, 17 logic, first-order, 185 order-extension, 12 Lojasiewicz, S., 49, 111 order-isomorphism, 17 Lombardi, H., 110, 111, 201 order-preserving, 17 Lorenz, F., 79 ordered field, 7 Los' theorem, 39 - Archimedean, 8 - special case of, 40 -- embedding into R, 9, 28 - characteristic of an , 8 Mabe, L., 201 - history of, 28 Marshall, M., 137, 159 - maximal, 14 maximal ordered, 14 ordered group, 24 McEnerney, J ., 49 - Archimedean, 28 measure ordered integral domain, 29 - positive Borel, 152 ordered skew-field, 28 metric, 223 ordered subfield, 12, 21 Minkowski's Theorem, 137, 157 ordering Minkowski, H., 49, 50, 188, 250 - 7]1-,48 Index 263 - Archimedean, 8 - ab stract, 81, 88, 109 -- in addition, see ordering, -- generalized, 86 of R[Xl, ... ,X n]:= R[X 1 , •• • ,Xn ]l! - due to Krivine, 110 - extension of an , 12-16 - weak (for modules of level 2m), 164 - - Archimedean, 26n - weak (for quadratic modules), 116 -- odd-degree , 13-14 power series, formal, 25, 28 -- quadratic, 13 Powers, V., 201 -- transcendental, 14 prenex definition, see - field, 3, 7 definition, prenex - group, 24 prenex statement, 35 - history of, 28 preordering - induced, 12 - Archimedean, 4, 116 - lexicographic, 134 - ofR[Xl , ... ,Xn ] - linear, 1, 7 -- canonical, 1,4 - of level 2m, 177 - of R[Xl, ... , x n] := R[Xl, ... ,Xn]1I - of V -- Archimedean, 117 -- canonical, 190, 191 - of C(x~ax , R) - of Vim -- canonical, 4 -- canonical, 26 - of a field, 10 - of v P - of a ring , 1, 81 - - canonical, 190 - of level 2m , 161 - of Q(Xd, 9 - - on K, extends to an ordering of - of R(XI), 8-9 level 2 if m odd, 175 - of level 2m , 247 - support of a, 81 - ofR[xl , . . . ,xn ] :=R[Xl , ... , Xn l/I prepositive cone - - Archimedean, 117 - of a field, 10, 29 - support of an , 81 - of a ring , 1, 81-82 - unique, 14, 25 - in addition, see preordering orthogonal bas is, 54 preprime orthogonal sum, see - Archimedean, 130-133 quadratic form Prestel, A., 29, 49, 79, 110, 111, 137, 158, 158n, 159, 178, 201, 230n, 245 p-adic numbers, 226 PrieB-Crampe, 28, 29 p-adic valuation, see valuation, p-ad ic prime ideal P-module, see module, P - characteristic of a, 61 P-semiordering, see primit ive polynomial, 218 semiordering, P principal ultrafilter, see Pfister, A., 69 ultrafilter, principal - 2n-bound, 68, 69, 80, 183 product, see - form , see form of degree 2m , an d quadratic form quadratic form projection theorem, 33 - Local-Global Principle, 53, 66, 74, 79 - over A, 34 P6lya's Theorem, 131, 136, 137 psd,49 positive Borel measure, see measure - over R, 36 positive cone Putinar, M., 137, 159 - of a field, 10 Pythagoras number, 80, 179 - of a ring, 81-85 - 2mth, 178 - in addition , see ordering - in addition, see p(A) positive semidefinite, 2, 49 Pythagorean field, 78 - on Sper A, 2 - over R, 36, 74 quadratic form Positivstellensatz , 90, 109, 110, 136, -D, the value set of a, 69 201, 247, 249, 250, 254 - anisotropic, 55 264 Index - associated symmetric bilinear form, - existence, 16 54 - notation for, 21 - definite w.r.t. a semiordering, 139 - uniqueness, 21 - definition of a, 53 Recio, T., 49, 249 - diagonal, 54 recursively enumerable, 185 - dimension of a, 53 refinement, see valuation, ring - equivalent, 54 regular, see form of degree 2m, and - hyperbolic, 56 quadratic form - indefinite over a field regular part, see form of degree 2m, -- totally, 73, 76n and quadratic form - indefinite w.r.t. a semiordering, 139 relatively - indefinite w.r.t. an ordering, 62 - algebraically closed, 23 - isotropic, 55 - complete, see valued field -- weakly, 74, 79, 139, 158 - separably closed, 228 - matrix of a, 53 Representation Theorem - orthogonal sum of, 56 - for modules of level 2m (Jacobi), 170 - Pfister form, 69 - for preorderings, 4, 109, 121 - product of, 59 -- Stone's, 137 - regular, 56 - for preprimes, 132 - regular part of a, 141, 142 - for quadratic modules (Jacobi), 127 - representing a, 55 - history of the, 136-137 - round, 68 representing a, see quadratic form - signature of a, 62, 63 residue - similar, 59 - degree, see valuation, extension of a - similarity class of a, 60 - field, see valuation - total signature of a, 65 - map, see valuation - totally indefinite, see quadratic form, Ressel, P., 155, 159 indefinite Reznick , B., 178, 201 - weakly isotropic, see quadratic form, Ribenboim, P., 245 isotropic Riesz Representation Theorem, 153 - in addition, see form of degree 2m ring quadratic module, see - local, 167, 203, 253 module, quadratic - of fractions, 108 quadratic system of representatives, 28 - real, 82 quantifier-free definition, see - semireal, 1, 82 definition, quantifier-free Risler, J .-J. , 110 roots quasi-compact, 85 - bound on, 19 quotient field, see field, of fractions Rosenberg, A., 79 quotient group, 25 round, see quadratic form Roy, M.-F., 48, 49, 109, 110 radical of an ideal, 65 Rudin, W., 153 - real, 88 Ruiz , J. , 48, 49, 110 ramification index, see valuation, extension of a SAP, 78, 79 rank 1, see valuation - of a real function field in one variable, rational rank, 240 72 real, see algebra, algebraic geom - valuation theoretic characterization etry, field, function field, ideal, of, 79, 158n Nullstellensatz, radical, ring, or saturation, 40, 49, 94, 110 spectrum scale,50n real algebraic numbers, 42 Scharlau, W., 159 real closed, 14, 29 Scheiderer, C., 29,49, 79, 80 real closure, VI, 16 Schilling, O.F.G., 245 Index 265 Schmid, J ., 178, 201 - - of IR[X1 , ,Xn ], 81, 101, 102, 110 Schmiidgen's Theorem, 4, 5, 123, 136, -- ofIR[Xl, , X n]/I, 106-107 13~ 139, 144, 150, 159, 185, 192, -- of Z, 84 196,254 - - of a field, 84 Schmiidgen, K., 137, 159 -- topologies on, see Schreier, 0 ., 14, 21, 29, 79, 109 topology on Sper A Schwartz, N., 137 -- in addition, see Schweighofer, M., VI, 137, 201 space of orderings semialgebraic - Zariski , 61 - definition, see -- of Z , 84 definition, semialgebraic - - of a field, 84 - function, 94n standard part, 103, 189 - set, 31-36 statement - - basic closed, 32 - prenex,35 - - basic open, 32 Stengle, G., 110, 201 -- bounded and basic closed, see Stone-Weierstrafi Theorem, 123, 134, WR(h 1 , .. ., hs ), WR(h) ~ R(n) 159 semiordering, 5, 79, 113, 115, 137 Strong Approximation Property, see - T- (T a preordering), 114 SAP - - support of a, 115 Sturm - of IR[Xl' ... , x n ] := IR[X1 , ••• ,Xn ]/I - sequence, 18 -- Archimedean, 117 - theorem, 18 - of level 2m, 163 subfield -- Archimedean, 164 - dense , see dense subfield - - determines a valuation ring , 166 - ordered, see ordered subfield -- history of, 177 subgroup, convex , see convex subgroup -- support of a, 163 subring, convex, see convex subring semireal ring, 1, 82 subsemiring, 86n separable sum - closure, 213n - of 2mth powers, 161 - polynomials, 218 - of squares, 11 Serre, J.P., 29 -- length of, 179 seventeenth problem, see Hilbert, D. support Siegel, C.L., 29n - of a subset of a ring , 81 signature, see quadratic form - in particular, see module, similar, see quadratic form (pre)ordering, preprime, or similarity classes, see semiordering quadratic form Sylow subgroup, 15 space of orderings, 65 - weak topology on, 66 Tarski, A., 49 - in addition, see spectrum, real - elimination of quantifiers, 33, 49, 79 specialization, 2, 3, 31, 81, 89, 103 -- generalized, 34 - closure under, 105 - Transfer Principle, 2, 3, 23, 31, 35, spectral topology, see topology 49, 110 spectrum topology - real, VI, 2, 4, 84, 109 - on IR(n) -- embeds in {O, 1}A, 107 -- discrete, 104 -- maximal, 3 -- interval, 104 maximal: compactness of, 107 - on R (n) maximal: found by Krivine, 109 -- interval, 32 of R[Xd, 84 - on Semi-Sper A of IR[X1, ... , X n ]: ~ space of -- constructible, 119 n-types, 110 -- spectral, 119 266 Index - on Sper A -- Chevalley's Theorem, 167, 205 -- constructible (or "canonical"), 4, 85 -- coarsening of a, 230 -- spectral, 85 -- composition, 232 - product (preserves Hausdorffness, -- dependence class of a, 233 (quasi-)compactness),107 -- dependent, 233 torsion element, 66 -- determined by a semiordering of n - 2 _, 65 level 2m, 166 torsion subgroup, 66 -- determined by a valuation, 204 total signature, see quadratic form -- independent, 234 totally indefinite, see quadratic form -- refinement of a, 231 totally isotropic, see -- trivial, 12, 203 quadratic form - in addit ion, see valued field totally positive, 29 value group, 27, 204 totally real , 50 valued field transcendence degree, 96 - algebraically maximal, 221 transcendental, see valuation, extension - complete, 228 of a -- relatively, 228 Transfer Principle, see Tarski, A. - completion of a, 228 Tressl , M., 110 - finitely ramified, 221 Tsen-Lang Theorem, 68 - Henselian, 27, 216 n-types, space of, ~ SperJR[Xl, ... , XnJ, - Henselization (or Henselian closure) 110 of a, 159, 213 -- characterization of the, 217, 222 ultrafilter, 37 - rank-1 - principal, 37 -- complete, 224 ultrapower, 38 - relatively complete, see ultraproduct, 36-41 valued field, complete uniquely orderable, see - in addit ion, see valuation ordering, unique van den Dries, see Dries variety, 157 valuation, 27, 204 - affine, 4 - discrete, 242 - compact real, 152 - extension of a, 206 Voevodsky's theorem, 80 algebraic, 207-213 Conjugation Theorem, 212 decomposition field of a, 213 Ware, R., 79 decomposition group of a, 213 weak Positivstellensatz, see existence of an, 207 Positivstellensatz, weak -- Gauss, 218 weakly isotropic, weakly 2m-isotropic, -- immediate, 217 see form of degree 2m, and quadratic -- ramification index e of an , 207 form -- residue degree f of an, 207 web site, VI -- transcendental, 236-242 well ordered, 25n - Krull, 29 Witt, E., 79 - of rank 1, 223, 242 - Cancellation Theorem, 57 - p-adic, 204 - Decomposition Theorem, 57 - residue field of a, 26, 203 - Local-Global Principle, 68, 73-74, - residue map of a, 27 76,79, 140 - ring, 12, 26, 203 - ring, VI, 53, 60, 79, 247, 250-252 -- Approximation Theorem, 234 -- fundamental ideal of the, 60 -- canonical, 175, 194 -- reduced, 66, 79, 110 canonical, see also 0 ~ JR" , -- torsion subgroup of the, 66 O(B, ~), O(P) , etc. Wormann, T., 136, 159, 177 Index 267 Zariski spectrum, see Ziegler, M., 111, 230n, 245 spectrum, Zariski zero-divisors, 108 Springer Monographs in Mathematics This series publishes advanced monographs giving well-written presentations of the "state-of-the-art» in fields of mathematical research that have acquired the maturityneeded for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for manyyears. Besides the current state of knowledge in its field, an SMM volume should also describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research. Abhyankar, S.S. Resolution of Singularities of Embedded Algebraic Surfaces 2nd enlarged ed 1998 Armitage, D.H.; Gardiner, S.J. Classical Potential Theory 2001 Arnold, L. Random Dynamical Systems 1998 Aubin, T. Some Nonlinear Problems in Riemannian Geometry 1998 Bang-Jensen, J.; Gutin, G. Digraphs 2001 Baues, H.-J. Combinatorial Foundation of Homology and Homotopy 1999 Brown, K.S.Buildings 1st ed.1998. 3rd printing2000 Cherry, W.;Ye,Z. Nevanlinna's Theory of Value Distribution 2001 Ching, W.K.lterative Methods for Queuing and Manufacturing Systems 2001 Crabb, M.C.; James, l.M. 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High-dimensional Knot Theory 1998 Ribenbolm, P.The Theory of Classical Valuations 1999 Rowe, E.G.P. Geometrical Physics in Minkowski Spacetime 2001 Rudyak, Y.B. On Thorn Spectra, Orientability and Cobordism 1998 Serre, J-P. Complex Semisimple Lie Algebras 2001 (reprint of first ed, 1987) Serre, J-P.Local Algebra 2000 Springer, T.A.; Veldkamp, F.D.Octonions, Jordan Algebras, and Exceptional Groups 2000 Sznitman, A.-S. Brownian Motion, Obstacles and Random Media 1998 Ustunel, A.-S.; Zakai, M. Transformation of Measure on Wiener Space 2000