Some Results from Theory

A.l Solvable Groups

Definition A.1.1. A for a group G is a sequence of sub­ groups G = Go2Gi 2...GA: = {1} with G/ a normal of G/_i for each /. A group G is solvable if it has a composition series with each quotient Gi/Gi-i abehan. o Example A.1.2. (1) Every abeUan group G is solvable (as we may choose Go = GandGi = {1}). (2) If G is the group of Lemma 4.6.1, G = (a, r | a^ = 1, r^~^ = 1, rar"^ =

(1) Gi-i/Gi is solvable for each i. (2) Gi-i/Gi is abelianfor each i. (3) Gi-i/Gi is cyclic for each i. (4) Gi-i/Gi is cyclic of prime order for each i.

Proof If Gi-i/Gi is solvable then we may "refine" the original sequence to G 2 • • 2 Gi-i = Ho 2 ^1 2 • • • 2 ^m = G/ 2 • • 2 G^ = {1} with Hj normal in Hj-i and Hj-i/Hj abelian; similarly we may refine a sequence with abelian quotients to one with cyclic quotients and one with cyclic quotients to one with quotients cyclic of prime order. On the other hand, if Gt-i/Gt is cyclic of prime order it is certainly solvable. n

Lemma A.1.4. (1) Let G be a and let H be a subgroup of G. Then H is a solvable group, (2) Let G be a solvable group and let N be a normal subgroup ofG. Then G/N is a solvable group, (3) Let G be a group and let N be a normal subgroup ofG.IfN and G/N are solvable, then G is solvable.

Proof Consider G = Go 2 Gi 2 • • • 2 G^ c {1}. Then H = HQ ^ Hi ^ • • • 2 ^A: 2 {1} where Hi = H D G/. It is easy to check that Hi is a normal subgroup of Hi-I, and then by standard theorems of group theory

Hi_,/Hi = (Hn Gi.i)/H n Gi = (Hn Gi-i)/(H n G,_i) n G, = (//nG,_i)G,/G,CG,_i/G, is isomorphic to a subgroup of an so it is abelian. (2) Let 7t:G-^ G/N = g be the quotient map and consider Q = QQ :D ... 2 e^ = {1} where Qi = n{Gi), Then Qi = GiN/N = Gi/Gi n N, so

7r(G,_i)/7r(G,) = (Gi.iN/N)/iGiN/N) = (Gi-i/Gi-inN)/(Gi/GinN)

is isomorphic to a quotient of the abelian group Gi-i/Gi so it is abelian. (3) Let TT: G -^ G/N = Q, Lei N = No ^ -" ^ Nj = {1} and G = Go 2 • • • 2 2A: = {1} be as in Definition A. 1.1. Then the sequence of

G = 7c-\Qo) 2 7t-\Qi) 2 • • • 2 yr'^Qk) = iVo 2 M 2 • • • 2 iV^ = {1}

shows that G is solvable. n A. 1 Solvable Groups 171

So far, we have considered abstract solvable groups. Hov^ever, when we consider solvable subgroups of symmetric groups, we can get very precise descriptions. Lemma A.1.5. Let G be a subgroup of Sp of order divisible by p, p a prime. Then any nontrivial normal subgroup KofG has order divisible by p. Proof Since G has order divisible by p, it must contain an element of order p, which must be a /?-cycle. Regard Sp as operating on T = {1,..., p]. Write T = Ti U • • • U T^, a decomposition onto orbits of K, In other words, /, y, G Tm for some m if and only if there is some x e K with x{i) = j. Now for any i, j e T there is a a e G with or(/) = j. This implies that the action of G permutes {Ti,..., T^} so in particular each T^ has the same cardinality. This cardinality then divides p, so must be 1 or /?, and cannot be 1 as ^ is nontrivial. Thus we see that K operates transitively on 7, so has order divisible by p (as the subgroup of K fixing 1, say, has index p). D

Lenvma A.1.6. Let G be a solvable subgroup of Sp of order divisible by p, p a prime. Then G contains a unique subgroup N of order p.

Proof Let G = Go D Gi D • • D Gjt = {1} as in Definition A.LL We prove the lenmia by induction on k. \fk = 1, G = Go D Gi = {1} and G is solvable, so G is abelian, and hence has a unique p-Sylow subgroup N, of order p. Suppose the lemma is true for A: — L Then Gi is a normal subgroup of G, so applying Lemma A.L5 with A' = Gi we see that G\ has order divisible by p, and then by the inductive hypothesis that G\ contains a unique subgroup N of order p. Now let A^' be any subgroup of G of order p. Then N' and N are both p-Sylow subgroups, so are conjugate. Let A^' = aNa~^. As N c Gi, A^' = aNcr~^ c aGia~^ = Gi as Gi is normal in G, so N^ = N, as required. D

Let a group G operate on a set T. This action is effective if the only ele­ ment a or G with a(t) = ^ for all ^ G T is a = id. Let a group Gi operate on a set Ti, and a group G2 operate on a set T2. These operations are permutation isomorphic if there is an isomorphism of groups xlr: Gi ^ G2 and an isomorphism of sets (i.e., a bijection) *: Ti -> T2 with *(a(0) = ^(or)(vl/(0) for every a eGut e Ti. Proposition A.1.7. Let G be a solvable group operating effectively and tran­ sitively on a set T of cardinality p, p a prime. Then there is a subgroup H of F* such that this operation is permutation isomorphic to 172 A Some Results from Group Theory

^^ = l[oi]' ^e^'^eFp} operating on

=![;]'•• €F,) by left multiplication.

Proof. As we have observed, in this situation G has order divisible by p. (Let ^ € r be arbitrary and consider [a e G \ a{t) = t]. Since G acts transitively on T, this subgroup of G has index p.) Since G operates effectively on T, G is isomorphic to a subgroup of the 5^,, so, by Lemma A. 1.6, G contains a unique subgroup N of order p, which is generated by a p-cycle. Reordering, if necessary, we have that A^ and T are permutation isomorphic to j L . | n e F^ I operating on

I I / € Fp I by left multiplication, as a suitable generator v of N takes

mod p. For simplicity of notation, we simply identify N with thiMV]s group and T with this set. Since A^ is the unique subgroup of order p of G, it is certainly a normal subgroup of G. Let a € G be arbitrary. Then ava~^ = v^ for some h ^ 0. Let Then, for any /,

so ^([;])=[o"][i] and hence

heV;, neFpY

Thus, if H is the subgroup of F* given by A.2 /7-Groups 173 A.=|*^ni[*„?]^4

then G = GH as claimed. n

Corollary A.1.8. Let G be a subgroup of Spy p a prime, of order divisible by p. The following are equivalent: (1) G is solvable. (2) The order ofG divides p(p — 1). (3) The order ofG is at most p(p — I).

Proof (1) implies (2): If G is solvable, then, by Proposition A. 1.7, G = GH for some H and then \G\ divides p(p — I). (2) implies (3): Trivial. (3) implies (1): G has a /7-Sylow subgroup A^, and N is unique, as if G had another /?-Sylow subgroup N\ then G would have order at least p^. Then the last part of the proof of Proposition A. 1.7 shows that G = GH for some H, But GH ::> N D {1} with A^ normal in G. N is abeUan and GH/N is isomorphic to H which is abelian, so G is solvable. n

A.2 p-Groups

Throughout this section, p denotes a prime.

Definition A.2.1. A group G is a /? — group if |G| is a power of /?. o

The center Z(G) of a group G is defined by Z(G) = {a e G \ at = ra for every r e G}.

Lemma A.2.2. Let G be a p-group. Then G has a nontrivial center (i.e., Z{G) ^ {1}).

Proof Write G = Ci U • U Q, a union of conjugacy classes. If at e Ct, then IQI = [G : Hi] where Hi = {r e G \ ra/t-^ = a,}. If a, e Z(G), then Q = {a/} and |C/| = 1 (and Hi = G). If a/ ^ Z(G), then /f/ is a proper subgroup of G, so IC/1 = [G^ : Hi ] is a positive power of p. Of course | G | is a positive power of p. But one of the conjugacy classes, say Ci, consists of {id} alone, and |Ci| = 1 is not divisible by p. Now \G\ = \Ci\-\ h |Q|. Hence there must exist other classes Ci with |Q | not divisible by p, and hence with \Ci I = 1, and if Q = {a,} then a, e Z(G). D 174 A Some Results from Group Theory

Corollary A.2.3. Let G be a p-groupy \G\ = p^. Then there is a sequence of subgroups G = Go D Gi D--DG„ = {1} with Gt a normal subgroup ofG of index p\ for each i = 1, ..., n.

Proof By induction on n. If n = 1, the result is trivial. Now suppose the result is true for all groups of order p^~^, and let G be an arbitrary group of order p^. By Lemma A.2.2, the center Z(G) of G is nontrivial. Since |Z(G)| divides |G|, |Z(G)| is also a power of p, so, in particular, Z(G) contains an element of order p. The cyclic subgroup H of G generated by that element is a subgroup of G of order p, and, since H c Z(G), H is a normal subgroup of G. Let Q = G/H he the , and let TT: G ^^ 2 be the canonical projection. Now 2 is a group of order p^~^, sohy the inductive hypothesis there is a sequence of subgroups Q = Qo D Qi D - " D Qn-i = {1} with Qt a normal subgroup of Q of index p^ for each / = 1,..., n — 1. Let Gt = 7t~^(Qi) for / = 1,..., n - 1 and G„ = {1}. Then G = Go D Gi D • • D G„ = {1} form a sequence as claimed. (Clearly Gt has index p^. Also, Gt is a normal subgroup of G as if go e Gt and g e G, then ggog~^ e Gt as n(ggog~^) = ^(g)^(go)(^(8))~^ ^ Qi as Qi is a normal subgroup of Q.) n

Lemma A.2.4. Let G be a p-group. Then G is solvable.

Proof By Corollary A.2.3, there is a sequence of subgroups G = Go D Gi Z) • • • D G„ = {1} with Gi a normal subgroup of G, and hence certainly a normal subgroup of G^_i, for each /. Furthermore, since [G : Gt] = p^ for each /, [Gi-i : Gi] = p and hence |G^_i/G/| = p, and so G/_i/G/ is cyclic of order p and, in particular, is abelian, for each /. Hence, by Definition A. 1.1, G is solvable. n

A.3 Symmetric and Alternating Groups

In this section we prove several results about symmetric and alternating groups. We let 5„ be the symmetric group on {1,..., n} and we let A„ c 5„ be the alternating group. We regard the elements of 5„ as functions on this set and recall that functions are composed by applying the rightmost function first. Thus, for example, (1 2) (1 3) = (1 3 2). We recall that every element of Sn can be written as a product of disjoint cycles, and that disjoint cycles conmiute. The first two results we prove allow us to conclude that certain subgroups of Sn are in fact equal to 5^. A. 3 Symmetric and Alternating Groups 175

Lemma A.3.1. Let p be a prime and let G be a transitive subgroup of Sp that contains a transposition. Then G = Sp. Proof. By renumbering if necessary, we may assume the transposition is r = (12). Since G acts transitively on {1,..., p}, it has order divisible by p and hence it has an element OTQ of order p. Since p is a prime, ao is a p-cycle. Thus there is some power a = a^ of (JQ with a (I) = 2. By renumbering if necessary, we may then assume a = (1 2 - - - p). Now direct calculation shows that

aJra-^ =(12-.. pY{I 2)(1 2 • • • p)"^' = ((j + 1) {j + 2)) for 7 = 0,...,/? — 2. Direct calculation then shows that

(2 3)(12)(2 3) = (13), (3 4)(13)(3 4) = (14),

i(p - 1) p){l (p - imip -l)p) = (1 p) and furthermore that

(l/:)(l7)(l/:) = 0/:)forany;7^/:.

Thus G contains every transposition. But Sp is generated by transpositions, so G = Sp. n

Lemma A.3.2. Let G be a transitive subgroup of Sn that contains an (n — l)- cycle and a transposition. Then G = Sn. Proof. Let p be the n — 1-cycle. By renumbering if necessary, we may assume that p = (12 " ' (n — I)). Then the transposition is r = (/ j) for some / and j. Since G operates transitively on {I,... ,n}, there is an element y of G with y(j) = n. Then y{i) = k for some k. Direct calculation shows that

yry~^ = (k n) = Xk and then that p^XkP~^ = ((k + i) n) = Xk+t, where A: + / is taken mod n — \. Finally, direct calculation shows that

XiXjxr^ = (i j) for any / ^ j.

Thus G contains every transposition. But Sn is generated by transpositions, so G = 5„. D 176 A Some Results from Group Theory

Next we show some results on the structure of A„ and 5„. We first need the following technical lemma.

Lemma A.3.3. IfG is a normal subgroup of An that contains a S-cycle^ then

Proof. We shall first show that if G contains a single 3-cycle, then it contains every 3-cycle. Note that if G contains a 3-cycle (/ j k), it contains {i j k)~^ = {kji). By renumbering if necessary, we may assume that G contains the 3-cycle (1 2 3). Then for each / > 3 (observing that an element of order 2 is its own inverse) G also contains the elements

((12)(3 0)(3 2 1)((12)(3 0) = (12 0, ((13)(2 0)(12 3)((13)(2/))=:(13 0, ((2 3)(10)(3 2 1)((2 3)(10) = (2 3/); then for each distinct i, j > 3, G also contains the elements

(12 7)(2/l) = (l/7), (2 3 7)(3/2) = (2/7), (3 1 7)(1 / 3) = (3 / 7); finally, for each distinct i, j,k > 3, G also contains the elements

(kil)(lij) = (ijk), so G contains all 3-cycles, as claimed. But we now claim that any element of A„ can be written as a product of 3- cycles. Since any element of An can be written as a product of an even number of transpositions, to show this it suffices to show that the product of any two transpositions can be written as a product of 3-cycles, and we see this from

(/ j)(ik) = (ikj), (ij)(kl) = (ikj)(iki), completing the proof. n

Theorem A.3.4. For n > 5, A„ is a (Le,y it has no normal sub­ groups other than An and {1}). A. 3 Symmetric and Alternating Groups 177

Proof. Let n > 5 and let G 7^ {1} be a normal subgroup of A„. We will show

By Lemma A.3.3, it suffices to show that G contains a 3-cycle, so that is what we shall do. Let a € G, or 7^ L Then a has order m > L Let /? be a prime dividing m and let y = cr^^^^ Then y has order p. Now y can be written as a product of disjoint cycles, y = y^ • • • y^, and since y has order /?, each of yi,..., yyt is a p-cycle. Now we must consider several cases: Case \: p > 5: By renumbering if necessary, we may assume that yi = (1 2 3 4 5 6 7 ... /7). Let n = (1 2)(3 4). Then fix = xmT7^ = (14 3 5 6 7 • • • /? 2) and yi^f ^ = (13542). Letting ^ = Xiyr~^ and using the fact that disjoint cycles commute, it easily follows that y^~^ = (13 5 4 2). Let T2 = (1 3)(2 4). Then r2(l 3 5 4 2)x2^ = (15243) and (13 5 42)(1 5 24 3) = (145). Case II: /? = 3: If A: = 1, i.e., there is only a single 3-cycle, there is nothing to do, so assume ^ > 2. By renumbering if necessary, we may assume that yi = (1 2 3) and y2 = (4 5 6). Let yi2 = yiy2 = (1 2 3)(4 5 6). Let n = (1 2)(3 4). Then Sn = xiynt^^ = (14 2)(3 5 6) and yi25-^ = (13425). Letting 8 = riyrf \ it then follows, as in Case I, that yS~^ = (13 4 2 5). Thus G contains a 5-cycle and we are back in Case I. Case III: p = 2: Since we are in A„, A: is even, so A: > 2. By renumbering if necessary, we may assume that yi = (12) and y2 = (3 4). Let yi2 = YiYi = (1 2)(3 4). Let p = (1 2 3). Then ^n = pynP'^ = d 4)(2 3) and yi2^^2^ = (1 3)(2 4). Letting ^ = pyp~^, it then follows, as in Case I, that yf-^ = 6> = (1 3)(2 4). Let rs = (1 3)(2 5). (Note here is where we need n > 5.) Then tser-^ = 6' = (I 3)(4 5). But then 66' = (2 4 5). D Theorem A.3.5. For n > 5, the only normal subgroups of Sn are 5„, A„, and {!}. Proof Let n > 5 and let G 7^ {1} be a normal subgroup of 5^. We will show G ^ An. Since An is a subgroup of Sn of index 2, this shows that G = An or Sn. The proof of this is almost identical to the proof of Theorem A.3.4. We begin in the same way. Let a e G,a ^ I, Then a has order m > 1. Let p be a prime dividing m and let y = a"^^^. Then y has order p. If we are in Case I, Case II, or Case III with A: > 2 of the proof of Theorem A.3.4, we conclude in exactly the same way that G 2 A„. This leaves only the case where p = 2 and k = I, i.e., where y is a single transposition. But all transpositions in 5„ are conjugate, and Sn is generated by transpositions, so in this case G = S^. D 178 A Some Results from Group Theory

Corollary A.3.6. For n > 5, neither A„ nor Sn is solvable.

Proof. Let n > 5. By the definition of a solvable group (Definition A. 1.1), if G = A„ or Sn were solvable, it would have a composition series with abelian quotients. But by Theorem A.3.4, the only possible composition series for A„ is A„ C {1}, and by Theorem A.3.5, the only possible composition series for Sn are Sn C {1} and Sn^ AnZ:> {1}, and A„ is not abelian. n B A Lemma on Constructing Fields

In Section 2.2 we showed: If fiX) e F[X] is an irreducible , then ^[X]/(f(X)) is a . We did so by using arguments particular to polynomi­ als. In this appendix we shall show that this result follow from more general results in ring theory. In particular, in this section we show how to construct fields from integral domains.

Definition B.1.1. Let R be an integral . (1) An ideal / C /? is a maximal ideal if / c y c /?, 7 an ideal, implies J = IorJ = R, (2) An ideal / c /? is a prime ideal if ab e /, with a,b e /?, implies aelorbeL o

Lemma B.1.2. (1) Let R be an integral domain. Then every maximal ideal is prime. (2) Let R be a PID (principal ideal domain). Then every prime ideal is maximal.

Proof. (1) Let / be a maximal ideal and let ab = i e I. If a e /, there is nothing to prove. Otherwise, let / = {a, I) (the ideal generated by a and /). Then I C J and / is maximal, so / = /?. In particular, 1 € /. Then xa + /' = 1 for some x e R and i^ e I. Then

b = l-b = ixa + i')b = X{ab) + i'b = iX + i'b e L

(2) Suppose / is a prime ideal and / is an ideal with I c. J, Then I = {p) for some p e R and / = {q) for some q e R. Since I c.J,peJ so p = qr for some r e R. Since / is prime, we conclude that: (a) ^ e / in which case J = {q) c. I so J = I,or 180 B A Lemma on Constructing Fields

(b) r € / so r = pr^ for some r' e R'm which case p = qr = qpr' = piqr') so 1 = qr^ and hence 1 € 7, so / = /?. D

Lemma B.1.3. Let R be an integral domain and I d R an ideal. Then I is maximal if and only if R/I is afield.

Proof Let n : R ^^ R/I be the canonical projection. First suppose / is maximal. Let a e R/I, a 7^ 0. Then a = 7t(a) for some a e R, a ^ I. Since / is maximal, R = {a, I), so 1 = jca + / for some x e R and some / e I. Then, if x = 7t(x),

1 = 7r(l) = n{xa + i) = xa + 0 = xd, so a is invertible. Now suppose / is not maximal, and let / c / C /?. Let j e J, j ^ I, and let j = n(j). We claim j is not invertible. For suppose jk = 1, 7, k e R/I, and let k = 7t(k), Then jk = I + i for some / € /, so 1 = jk — i and hence 1 G /, a contradiction. D

Corollary B.1.4. Let R be a PID and let I = (i) be the ideal generated by the element i of R, Then R/I is afield if and only ifi is irreducible.

Proof. In a PID, an element / is irreducible if and only if / is prime, and an ideal / = (/} is a prime ideal if and only if the element / is prime, so the corollary follows directly from Lemma B. 1.2 and Lemma B.1.3. n

Corollary B.1.5. L^^ f{X) e F[Z] be an irreducible polynomial. Then F[X]/{f{X)) is afield.

Proof. Since F[X] is a PID, this is a special case of Corollary B.1.4. n A Lemma from Elementary

In this appendix, we prove a lemma that is used in the proof of Theorem 4.7.1. Lemma C.1.1. Let p be a prime and let t be a positive . Then there are infinitely many primes congruent to \(moA.p^). Proof. We begin by recalling the following definition and facts: Let q be a prime and let a be an integer relatively prime to q. Then ord^(fl), the order of a(mod^), is defined to be the smallest positive integer x such that a^ = l(modqr). Then, for an arbitrary integer y, a^ = l(mod^) if and only if ord^(a) divides y. Also, since, by Fermat's Little Theorem, a^~^ = l(mod^), we have that ord^(a) divides q — I. We now proceed with the proof, which we divide into two cases: p = 2 and p odd. Case I: p = 2. For a nonnegative integer k, let Fk = 2^^ +1. Note that, if j < it, Fj = 2^' +1 divides 2^' -1 = (2^' )^'"' -1 = F)t - 2, so gcd(F^, Fk) = gcd(Fj, 2) = 1, i.e., {Fi, F2,...} are pairwise relatively prime. Thus, if qk is a prime factor of Fk, then {^1, ^2, • •} are all distinct. Now let ^ = qs for s > ^ — 1. By definition, q divides Fy, i.e., 2^' + 1 = O(mod^), 2^' = -l(mod^), and hence 2^""' = l(mod^). Thus ord^(2) divides 2^"*"^ but does not divide 2^, so ord^(2) = 2^"'"^ Then, as we have observed above, 2*+^ divides ^ — 1, so ^ = l(mod2^+^) and hence q = l(mod20. Thus we see that {qt-.i,qt,...} are distinct primes congruent to l(mod20. Case II: p odd. In this case we proceed by contradiction. Assume there are only finitely many primes = l(modp^), and let these bcqi,,.. ,qr. Let

a = 2qx'"qr, c = a^"', N = a^' - 1 = c^ - I, 182 C A Lemma from Elementary Number Theory

and write N = (c — 1)M where M = CP~^ -\ hi. Claim: c — \ and M are relatively prime. Proof of claim: M = CP-\ + • • • + 1 = (c^-^ - 1) + (c^-^ -!) + ••• + (c — 1) +p, and each term except possibly the last one is divisible by c — 1, so gcd(c — 1, M) = gcd(c — 1, p) = I or p. But a = 2(mod/?) so c = 2P'~\modp), By Fermat's Little Theorem, 2^"^ = l(modp), and p^~^ = l(modp — 1), so c = 2^ = 2(modp) and so c — 1 = l(mod/7). Thus p does not divide c — 1 and so gcd(c — I, M) = gcd(c — I, p) = I. Now let q be any prime dividing M. Then q divides A^ = c^ — 1, so cP = l(mod^), i.e., a^' = l(mod^). So ord^(a) divides p\ i.e., ord^(a) = 1, /7,..., p^~^ or /7^ But if ord^(fl) = 1, /?,..., or /?^~\ then a^' = l(mod^), i.e., c = l(mod^), so q divides c — 1, contradicting the fact that c — 1 and M are relatively prime. Hence ord^(a) = pK Then, as we have observed above, p^ divides q — I, i.e., q = l(modp^). But certainly q does not divides a, while each of qi,... ,qr does divide a, so q ^ qi,... ,qr, contradicting the hypothesis that qi,... ,qr are all the primes congruent to l(mod/?0- Hence there must be infinitely many such primes. D Index

17-gon 101 separable, 145 Dirichlet 29,119 Abel 95 discriminant 125 adjoining 17 division algorithm 11 algebraic integer 42 domain algebraic number 18, 20 Euclidean, 11 algebraic number field 20 integral, 10, 179 algebraically closed 155 principal ideal, 11, 179 algorithm for computing Galois groups duplicating the cube 99 124 algorithm for factoring 88 Eisenstein's Criterion 87 Artin 30 element Artin-Schreier extension 80 algebraic, 18, 21 inseparable, 25 basis irreducible, 180 normal, 49 primitive, 47, 63, 64 character 29 purely inseparable, 143, 144 characteristic 9 separable, 25, 141 closure extension 9 algebraic, 151, 154 abelian,70, 118, 119 Galois, 148 algebraic, 18, 20, 21 normal, 148 composite, 166 separable, 143, 145 finite, 9 conjugate Galois, 27, 28, 32, 35, 58, 59, 61, 157, Galois, 27 158, 166, 167 inseparable, 25, 52 degree 9, 16, 17, 19, 23, 27, 58, 59, nonsimple, 66 113, 114,144 normal, 25, 26, 28, 32, 145, 147, 148, inseparable, 145 157 184 Index

purely inseparable, 143-145 cyclic, 12 quadratic, 104 Galois, 1, 26, 32, 36, 37,45, 55, 58, radical, 108 61,62,70-72,74,92,108,109, separable, 25, 26, 28, 51, 52, 139, 141, 111,113,116,118,119,122,123, 142, 157 164, 167 simple, 47, 63, 64 nonsolvable, 178 symmetric, 118, 119 /7-group, 173, 174 extension by radicals see radical permutation, 35 extension simple, 176 extensions solvable, 94, 169-171, 173, 174 conjugate, 34 symmetric, 119, 123, 171, 173-175, disjoint, 58-60, 92, 166 177 topological, 159 Fermat prime 100,101 Fermat's Little Theorem 10 Hubert's Theorem 90 77 field 7, 13, 180 algebraically closed, 151, 154 ideal composite, 16 maximal, 179, 180 cyclotomic, 89, 104, 108, 110, 119 prime, 179 extension, 14, 15 inseparability finite, 53, 54 degree of, 146 fixed, 61 inseparability, degree of 143 Kummer, 73, 108 Invariance of gcd under Field Extensions perfect, 26, 53 15 quadratic, 107, 108 splitting, 22, 24-26, 28, 147, 157, 158 Kronecker 14,88 Frobenius map 10, 53, 54, 139 Kronecker-Weber Theorem 119 FTGT see Fundamental Theorem of Lagrange Interpolation Formula 88 function Landau 91 synmietric, 45 Lindemann 99 Fundamental Theorem of Algebra 155 Fundamental Theorem of Galois Theory Mobius Inversion Formula 57, 93 2, 27, 32, 37, 55, 160 Fundamental Theorem of Infinite Galois n-powerless 71, 72, 110 Theory 160 Newton's identities 83 norm 76 Galois 95 normal basis 66, 69 Gauss 101, 108 Gauss's Lenmia 86 polynomial gcd see greatest common divisor cyclotomic, 87, 89, 90 greatest common divisor 11,15 elementary symmetric, 46 group inseparable, 25, 52 abelian, 118 irreducible, 56, 86-88, 90, 112,180 alternating, 174,176 minimum, 18, 19, 27, 62, 63, 76, 147 Index 185

primitive, 86 straightedge and compass 97, 99, 101 radical, 93, 108 separable, 25, 26, 28, 51, 52, 141, 142, Theorem on Natural Irrationalities 60, 157,158 154 symmetric, 45 topology polynomial ring 11 compact-open, 166 Krull, 159, 163, 164, 166 93 product, 166 trace 76 solvable by radicals 93-95 transformation, Hnear 19 not, 95, 96 trisecting the angle 99 splits 22 squaring the circle 99 van der Waerden 119