The Discriminant MAT4250 — Høst 2013

The discriminant MAT4250 — Høst 2013

The discriminant

First and raw version 0.1 — 23. september 2013 klokken 13:45 One of the most significant invariant of an algebraic number field is the discriminant. One is tempted to say, apart from the degree, the most fundamental invariant. In simplest case of a quadratic equation, the discriminant tells us the behavior of solution, and of course, even its square roots gives us the solutions. To some extent the same is true for cubic equations, and the higher the degree of an equation less the influence of the discriminant is, but it always plays an important role. For a field extension we shall define a discriminant for any basis. Of course this depends on the basis, but in a very perspicuous way. In the case of a number field, the ring of algebraic integers A is a free module over Z,andifthebasisesareconfined to be Z-basis for A,thediscriminantisanintegerindependentofthethebasis,isan invariant of the number field. The discriminant serves several purposes at. Its main main feature is that it tells us in which primes a number fields ramify, or more generally, in which prime ideals an extension ramifies. Additionally the discriminant is a valuable tool to find Z-basis for the ring A of algebraic integers in a number field. To describe A is in general a difficult task, and the discriminant is some times helpful. In relative situation, the situation is somehow more complicated, and the discriminant is well-defined just as an ideal.

The discriminant of a basis

The scenario is the usual one: A denotes a Dedekind ring with quotient field K,andL afinite,separableextensionofK. The integral closure of A in L is B.InSeparability we defined the trace form trL/K (xy).ItisasymmetricandquadraticformonL taking values in K,andsinceL is separable over K,itisnon-degenerateaftertheorem1 on page 10 of Separability. Let ↵1,...,↵n be a K-basis for L and denote it by .Onemayform—asonemaydo B for any quadratic form—the matrix M of the trace form by putting M =(trL/K (↵i↵j)). If ↵ and are the coordinate vectors relative to the basis of two elements ↵ and B B B of L,thevalueofthetraceformtrL/K (↵) is given as

t trKQ(↵)=↵ M B B We deﬁne the discriminant of the basis as the determinant of the matrix M. That B is =(↵1,...,↵n) = det(trK/Q(↵i↵j)). B The discriminant depends of course on the basis, but in a very clean way. Assume that 0 is another basis and denote by M 0 the matrix of the trace form in that basis. B —1— The discriminant MAT4250 — Høst 2013

If V denotes the transition matrix between the two basis—that is ↵ = V↵ for all ↵ B B0 —then t t t t ↵ M =(V↵ 0 ) M(V 0 )=↵ (V MV) 0 , B B B B B0 B t and as this holds for all ↵ and , one draws the conclusion that M 0 = V MV.Taking determinants, one obtains:

Lemma 1 Assume that and 0 are two K-basises for the field L with transitions B B matrix V (from 0 to ). Then the relation between the corresponding discriminants is B B given as: =(detV )2 B0 B An important observation is, that if is an integral basis, i.e., the basis elements B are all lying in B,thenthediscriminantbelongstoA.Indeed,alreadytheallthetraces trL/K (↵i↵j) lie in A. Example . — The case of a primitive basis. Assume that the field L has a primitive element x , i.e., L = K(x),andletn be the degree of x. Then L has the n 1 special basis 1,x,...,x formed by the n first powers of x. There is a nice expression for the discriminant of L over K in that particular basis, it equals the discriminant of the minimal polynomial of x as defined in Separability.Torealizethis,let!1,...,!n be the roots of the minimal polynomial of x in some big field ⌦.Inotherwords,the values the different embeddings of L in ⌦ take at x. The !k’s are the eigenvalues in ⌦ i i of the multiplication map ⇢x,andthereforetrL/K (x )= k !k. j 1 Let V =(! ) be the van der Monde matrix formed by the !0s. Then, forming i P i the product of V and its transposed, we get

t i 1 j 1 i 1 j 1 i+j 2 i 1 j 1 V V =(!j )(!i )=( !k !k )=( !k )=(trL/K (x x )), Xk Xk 2 n 1 and therefore 1,x,...,x =(detV ) . Combining this with xxxx, we obtain Proposition 1 Given a ﬁeld extension L = K(x) of degree n with primitive element x.Assumethatx has the minimal polynomial f(t) over K.Thenthediscriminantof n 1 the power basis 1,x...,x is equal to the discriminant of the polynomial f.Thatis n 1 f =1,x,...,x .

The case of extensions of a pid

Assume now that in the standard setup A is a principal ideal domain, and n =[L : K]. The two main cases that concerns us are the case of a number ﬁeld whose with ring of integers extends the principal ideal domain Z and the case that A is a dvr .IfA is

—2— The discriminant MAT4250 — Høst 2013

a pid ,anyﬁnitelygeneratedandtorsionfreemoduleoverA is a free module and thus has an A-basis. The transition matrix between two such Z-basises belongs to group Gl(n)A of invertible n n-matrices with coeﬃcients in A,anditsdeterminantbelongs ⇥ therefore to groups A⇤ of units.

The case of a number field In the all-important case of a number field the ring of integers A is a free Z-module, and the determinant of a transition matrix between two basises equals 1. The formula ± in lemma 1 above shows that the discriminant is independent of as long as we B B restrict to be a Z-basis of A. This number i called the discriminant of K and it is B denoted by K . Proposition 2 Assume that K is a number field and that A is the ring of integers in K.Thenthediscriminant is independent of the basis as long as is a Z-basis B B B for A.Itiscalledthe discriminant of K and denoted by K . Example . The discriminant of the quadratic field Q(pd) depends on the residue class of d mod 4. The discriminant is 4d in case d 1mod4and equals d if d 1 6⌘ ⌘ mod 4. e The discriminant and the ramified primes The most important application of the discriminant is that it detects ramification. In the case that A has a primitive element, say A = Z[⇠],thisisreasonable,sincethenaprimep ramifies exactly when the minimal polynomial f(x) of ⇠ has a multiple root modulo p,andthediscriminant of f(x) is made to detect this. However, in general A does not have a primitive element and this complicates the matter substantially, but still the discriminant behaves well:

Theorem 1 Assume that K is a number field with discriminant K .Thenaprimep is ramified in K if and only if p divides the discriminant K . Proof: There are three points, the first one being that the trace is functorial in the following sense. Let p Z be a prime and let a1,...,an be a Z basis for A. Then of 2 course a¯1,...,a¯n is a k(p) basis for A/pA. Clearly if one reduces the multiplication map ⇢a in A modulo p one obtains the multiplication map ⇢a¯ in A/pA.NowusinganZ-basis for A to compute trK/Q(a),oneseesthattheintegertrK/Q(a) reduces to trA/pA/k(p) (a) mod p.Applyingthistoallthecoefficientsofthematrix(trK/Q(aiaj)),oneobtainsthat det(trK/Q(aiaj)) reduces to det(tr(A/pA)/k(p)(¯aia¯j)) mod p,andasK =det(trK/Q(aiaj)), it follows that p divides K if and only if the trace form on the k(p)-algebra A/pA is degenerate. The second point is that trace form on A/pA is non-degenerate if and only if A/pA is a product of separable field extensions of k(p).Butask(p) is a finite field, all field extensions are separable, so this happens if and only if A/pA is a product of fields. That is, if and only if it has no nilpotent elements. ei The third and final point is that A/pA = 1 i s A/qi is without nilpotents if and   only if all the ei’s are equal to one, that is if and only if p is non-ramified. o Q

—3— The discriminant MAT4250 — Høst 2013

The discriminant and the ring of integers. The discriminant can sometimes be of great help to find the ring of integers in a number field. Assume we have at our disposal an integral basis of K,thatisaQ-basis for K contained in A.Assumethat B 0 is a Z-basis for A and let V be the transition matrix between these two basises. The B known basis will be a Z-basis for A if and only if V is invertible in Gl(Z)n,thatis, B if and only if det V is plus or minus one. But by lemma 1 above the two discriminant 2 and 0 differ by the factor (det V ) ,henceif is a square free integer,itmustbe B B B so that det V = 1,andourbasis is a Z-basis for A. ± B Proposition 3 Assume that K is a number field with ring of integers is A.Ifan integral basis for K has a square free determinant, then it is a Z-basis for A. Example . Let f(x)=x3 x 1,thenf(x) is irreducible (only possible roots are 1). ± Let K = Q(⇠) where ⇠ is one of the roots of f. The polynomial f(x) has discriminant 4( 1)3 27 12 = 23 which coincides with the diskriminant of the basis 1,⇠,⇠2. · This is obviously an integral basis, and 23 is square free. It follows that 1,⇠,⇠2 is a basis for the ring of integers. e

Problem 1. Let f(x)=x3 x +3.Showthatf(x) is irreducible over Q and if ⇠ denotes a root of f(x),ﬁndabasisfortheringofintegersK = Q(⇠). Hint: Compute the discriminant by the formula 4a3 27b2 (see Separability for this formula) X Problem 2. Let g(x)=x3 3x +9.Showthatg is irreducible and compute the discriminant of f.Let⌘ be a root of g(x).ShowthatQ(⌘)=Q(⇠)=K (where ⇠ is as in the previous problem). Show that Z[⌘] is not the ring of integers in K. Hint: 3/⇠ is a root of g(x). X

Problem 3. Assume that a1,...,an is an integral basis for A.Showthatifd = (a1,...,an),thendA a1Z + + anZ. X ✓ ··· Problem 4. Let 1,...,n be the different embeddings of the number field K into some big field ⌦.Let↵1,...,↵n be a basis for K over Q.Showthat 2 (↵1,...,↵n)=(det(i(↵j))) X

Problem 5. Let (i(↵j)) be the n n-matrix from problem 4.LetP = (↵1) ⇥ ⌧ even ⌧(1) · ⌧(n)(↵n) and N = ⌧(1)(↵1) ⌧(n)(↵n) where ⌧ runs through the permu- ·· ⌧ odd ··· P tations of 1,...,n .Showthatdet(aij)=P N,andthatbothP + N and PN are { } P invariant under the action of ⌧ that permutes the diﬀerent embeddings i. Then show that P and N both lie in Z. X

Problem 6. (Stickelberger). Let K be the discriminant of the number ﬁeld K.Show that K is a square modulo 4,henceK 0 or K 1 modulo 4. Hint: Write ⌘ ⌘ =(P N)2 =(P + N)2 4PN with a smart choice of P an N. X

—4— The discriminant MAT4250 — Høst 2013

The general case and the diskriminant ideal In the general case, there is no way of getting one element in A that plays the role of the discriminant as for a number field. There are two reasons for this. First of all, the ring B is not necessarily is a free A-module and we do have A-basises for B at our disposal. Secondly, even if B were a free A-module, the group A⇤ of units in A is in 2 general much more complicated than the one of Z and (A⇤) can be highly non-trivial. However one may define a discriminant ideal dB/A that does the job. It is defined as the ideal in A generated by (b1,...,bn) as the b1,...,bn run through all integral basises for L over K.Recallthatthismeansthatb1,...,bn is a basis for L contained in B. In case K is number field with ring of integers A,thediscriminantK is a slightly more subtle invariant than the discriminant ideal dA/Z.Ofcourse,K will be one of the two generators of dA/Z,thesubtilityliesinthefactthatoneofthemispreferredover the other. Just like for the discriminant, the interest of the discriminant deal lies in the fact that it detects ramification: Theorem 2 (Dedekind) Let A be a Dedekind ring with quotient field K and let B be the integral closure of A in a finite and separable extension L of K.Thenaprime p of A ramifies in B if and only if d p. B/A✓ Proof: The proof has three ingredient. The first one is that the discriminant ideal localizes well. That is

⇤ If S is a multiplicative system in A,wehavedBS /AS =(dB/A)S Clearly if ↵1,...,↵n is a K-basis for L contained in B,itisaK-basis for L contained in BS.Hencetheinclusion(d )S d .Ontheotherhand,if↵1,...,↵n is a K basis B/A ✓ BS /AS for L contained in the localization BS,thens↵1,...,s↵n is contained in B for a suitable 2n s S,anditisstillaK basis for L.Obviously(s↵1,...,s↵n)=s (↵1,...,↵n). 2 This shows that d (d )S. BS /AS ✓ B/A The second ingredient is that the ramification behavior of a prime ideal localizes. This somehow vague statement, means the following: ⇤ If p is a prime ideal i A,thenp ramifies in B if and only if pAp ramifies in Bp. Indeed, this is an immediate consequence of the isomorphism B/p B /pB and ' p p the fact that p ramifies (respectively pAp)ifandonlyifB/pB (resp. Bp/pBp)has non-trivial nilpotent elements. The third and final ingredient is that the theorem holds for the local rings Ap,in fact it holds whenever A is a pid : If A is a pid ,thenp ramifies if and only if d p. ⇤ B/A✓ Indeed, it follows from lemma 1 on page 2 that if ↵1,...,↵n is an integral basis for L, then (↵1,...,↵n) is a generator for the discriminant ideal dB/A.AndasB is a free A- module (any torsion free and finitely generated A-module is since A is, a pid )theproof of theorem 1 shows mutatis mutandi that p ramifies if and only if (↵1,...,↵n) / p. 2 o

—5— The discriminant MAT4250 — Høst 2013

An observation which is not aprioriclear, is that in the standard situation, only ﬁnitely many prime ideals ramify in B.

Versjon: Monday, September 23, 2013 1:45:34 PM

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