MATH5725 Lecture Notes∗ Typed By Charles Qin October 2007

1 Genesis Of Galois Theory

Definition 1.1 (Radical Extension). A field extension K/F is radical if there is a tower of ri field extensions F = F0 ⊆ F1 ⊆ F2 ⊆ ... ⊆ Fn = K where Fi+1 = Fi(αi), αi ∈ Fi for some + ri ∈ Z .

2 Splitting Fields

Proposition-Definition 2.1 (Field Homomorphism). A map of fields σ : F −→ F 0 is a field homomorphism if it is a ring homomorphism. We also have:

(i) σ is injective

(ii) σ[x]: F [x] −→ F 0[x] is a ring homomorphism where

n n X i X i σ[x]( fix ) = σ(fi)x i=1 i=1

Proposition 2.1. K = F [x]/hp(x)i is a field extension of F via composite ring homomor- phism F,−→ F [x] −→ F [x]/hp(x)i. Also K = F (α) where α = x + hp(x)i is a root of p(x).

Proposition 2.2. Let σ : F −→ F 0 be a field isomorphism (a bijective field homomorphism). Let p(x) ∈ F [x] be irreducible. Let α and α0 be roots of p(x) and (σp)(x) respectively (in appropriate field extensions). Then there is a field extensionσ ˜ : F (α) −→ F 0(α0) such that:

(i)σ ˜ extends σ, i.e.σ ˜|F = σ (ii)σ ˜(α) = α0

Definition 2.1 (Splitting Field). Let F be a field and f(x) ∈ F [x]. A field extension K/F is a splitting field for f(x) over F if:

∗The following notes were based on Dr Daniel Chan’s MATH5725 lectures in semester 2, 2007

1 (i) f(x) factors into linear polynomials over K

(ii) K = F (α1, α2, . . . , αn) where α1, α2, . . . , αn are the roots of f(x) in K

Note. Consider tower of field extension F ⊆ K ⊆ L and f(x) ∈ F [x]. If L is a splitting field for f(x) over F , then it is a splitting field for K. If K is generated by roots of f(x) then the converse also holds.

Theorem 2.1. Let F be a field and f(x) ∈ F [x]. Then there is a splitting field K of f(x) over F .

Theorem 2.2. Let σ : F −→ F 0 be a field isomorphism and f(x) ∈ F [x]. Suppose K, K0 are splitting fields for f(x) and (σf)(x) over F and F 0 respectively. Then there is an isomorphism of fieldsσ ˜ : K −→ K0 which extends σ. This is referred to as the uniqueness of splitting fields.

3 Algebraic Closure

Definition 3.1 (Algebraic Extension & Algebraic Closure). Some definitions.

(i) A field extension K/F is algebraic over F if every α ∈ K is algebraic over F

(ii) A field K is algebraically closed if the only algebraic extension of K is K itself

(iii) K is an algebraic closure of F if it is an algebraic field extension of F such that K is algebraically closed

Proposition 3.1. Let F ⊆ K ⊆ L be a tower field extensions then:

(i) [L : F ] = [L : K][K : F ]

(ii) L/F is algebraic if and only if both L/K and K/F are algebraic

(iii) Finite field extensions are algebraic

(iv) If K = F (α1, α2, . . . , αn) where α1, α2, . . . , αn are algebraic over F , then K is finite over F

Note. If α is transcendental, then [F (α): F ] = ∞ as irreducible polynomial with root α does not exist.

Lemma 3.1. Let F be a field. There exists a set S such that for any algebraic field extension K/F , |K| < |S|.

Remark. Either K is countable or |K| = |F |.

2 Definition 3.2 (Splitting Field). A field extension K/F is a splitting field for {fi(x)}i∈I over F if:

(i) every fi(x) factorises into linear factors over K

(ii) K is generated is the field generated by F and all the roots of all the fi(x)’s

Q Note. If I is finite, K is just the splitting field for i∈I fi(x).

Proposition 3.2. In field extension K/F , let {αj}j∈J ⊆ K. The subfield of K generated by F and the αj’s is:

(i) intersection of all the subfields of K containing F and all the αi’s

(ii) union of all F (αj1 , αj2 , . . . , αjn ) where {αj1 , αj2 , . . . , αjn } ⊆ J

Theorem 3.1. Let F be a field and {fi(x)}i∈I ⊆ F [x] − 0.

(i) There exists a splitting field K for {fi(x)}i∈I over F (ii) Suppose there is a field isomorphism σ : F −→ F 0 and K0 is a splitting field for 0 0 {(σfi)(x)}i∈I over F . There is a field isomorphismσ ˜ : K −→ K which extends σ.

4 Field Automorphisms

Proposition-Definition 4.1 (Field Homomorphism Over F ). Let K, K0 be field extensions of field F . We say a homomorphism σ : K −→ K0 fixes F or σ is a field homomorphism over F if σ(α) = α for any α ∈ F . Such homomorphisms are linear over F . If furthermore K = K0 and σ is an automorphism then we say σ is a field automorphism over F .

Proposition-Definition 4.2 (Galois Group). Let K/F be a field extension of F . Let G be the set of field automorphisms of K over F . Then G is a group when endowed with composition as group multiplication. It is called the Galois group of K/F and is denoted by Gal(K/F ).

Lemma 4.1. Let f(x) ∈ F [x]. K/F is a field extension and α ∈ K is a root of f(x). Let σ : K −→ K0 be a field homomorphism over F . Then σ(α) is also a root of f(x).

Remark. Any field homomorphism σ : F (α1, α2, . . . , αn) −→ K over F is determined by values σ(α1), σ(α2), . . . , σ(αn) since σ fixes F .

Corollary 4.1. Let K be a splitting field for f(x) ∈ F [x] over field F . Let α1, α2, . . . , αn be roots of f(x), so that K = F (α1, α2, . . . , αn). Then any σ ∈ G = Gal(K/F ) per- mutes the roots of α1, α2, . . . , αn and so gives an injective group homomorphism G,−→ ∼ Perm({α1, α2, . . . , αn}) = Sn.

Lemma 4.2. Let F be a field and f(x) ∈ F [x]. Let K the splitting field for f(x) over F . 0 Suppose α and α are roots of an irreducible factor f0(x) of f(x) over F . Then there is a 0 σ ∈ G = Gal(K/F ) such that σ(α) = α . In particular, G acts transitively on roots of f0(x).

3 5 Fixed Fields

Proposition-Definition 5.1 (Fixed Field). The fixed field of G in K is KG = {α ∈ K : σ(α) = α, for any σ ∈ G} and it is a subfield of K.

Definition 5.1 (Prime Maps). Let K/F be a field extension. G = K/F. We define two maps, both called prime. {subgroups of G} ←→ {intermediate fields of K/F }; H 7−→ H0 = KH ; 0 K0 7−→ K0 = Gal(K/K0).

Lemma 5.1. Let H1,H2,... denote subgroups of G and K1,K2,... the intermediate fields of K/F . Priming reverses inclusions:

0 0 (i) H1 ⊆ H2 =⇒ H1 ⊇ H2

0 0 (ii) K1 ⊆ K2 =⇒ K1 ⊇ K2

00 00 Lemma 5.2. H1 ⊆ H1 and K1 ⊆ K1 .

00 Definition 5.2 (Closed Subsets). We H1 is a closed subgroup of G if H1 = H1 and K1 is 00 00 00 a closed intermediate field of K/F of K1 = K1 . The closure of H1 and K1 are H1 and K1 respectively.

0 000 0 000 Lemma 5.3. H1 = H1 and K1 = K1 .

Theorem 5.1. Priming induces a well defined bijection {closed subgroups of G} ←→ {closed 0 H 0 intermediate fields of K/F }; H 7−→ H = K ; K0 7−→ K0 = Gal(K/K0).

6 Two Technical Results

0 0 Theorem 6.1. Let K1 ≤ K2 be intermediate fields. Then [K1 : K2] ≤ [K2 : K1] = n if n < ∞.

0 0 0 0 Lemma 6.1. Let σ, τ ∈ K1 then σK2 6= τK2 (distinct cosets) =⇒ σ(α) 6= τ(α), i.e. [K1 : 0 K2] ≤ number of distinct σ’s ≤ n.

0 Lemma 6.2. Let σ, τ ∈ G be such that σH1 = τH1. Then for any α ∈ H1, we have σ(α) = τ(α).

0 Remark. As a result, given any left coset C ⊆ H1 ⊆ G and α ∈ H1, we can unambiguously define C(α) = σ(α) where σ is any element of C.

0 0 Theorem 6.2. Let H1 ≤ H2 ≤ G. Suppose n = [H2 : H1] < ∞. Then [H1 : H2] ≤ [H2 : H1].

4 7 Galois Correspondence

Corollary 7.1. Two corresponding results.

(i) Let H1 ≤ H2 ≤ G. If H1 is closed and [H2 : H1] < ∞ then H2 is closed and [H2 : H1] = 0 0 [H1 : H2]

(ii) Given intermediate fields K1 ≤ K2 ≤ K. Supposed K1 is closed and [K2 : K1] < ∞ 0 0 then K2 is closed and [K2 : K1] = [K1 : K2]

Definition 7.1 (Galois Extension). An algebraic field extension K/F is Galois if F is closed, i.e. F = F 00 = KGal(K/F ).

Proposition 7.1. Let K/F be a finite field extension. (i) G = Gal(K/F ) is finite

(ii) K/F is Galois =⇒ [K : F ] = |G|

(iii) If |G| ≥ [K : F ] then K/F is Galois

Theorem 7.1 (Fundamental Theorem Of Galois Theory). Let K/F be a finite Galois exten- sion and G = Gal(K/F ). (i) There are inverse bijections {subgroups of G} ←→ {intermediate fields of K}; H 7−→ 0 H 0 H = K ; K0 7−→ K0 = Gal(K/K0)

0 (ii) For intermediate field K0, K/K0 is Galois with Galois group Gal(K/K0) = K0

Note. In fact, we can show that all intermediate fields of a Galois extension are closed. So are the subgroups of the corresponding Galois group.

Theorem 7.2. Let K be a field and G be a finite group of field automorphisms of K. If F = KG then K/F is Galois and the Galois group is G, i.e. K/KG is Galois with Galois group G.

8 Normality

Definition 8.1 (Normal Extension). K is a splitting field over F if it is the splitting field of some family of polynomials over F . We say also in this case that K/F is normal.

Proposition 8.1. Let K/F be a Galois extension with Galois group G. (i) Let α ∈ K and p(x) ∈ F [x] be its minimal polynomial. Then p(x) factorises over K

(ii) K is the splitting field of {fi} over F where fi ranges over the minimum polynomials of all α ∈ K

5 Lemma 8.1. Let K/F be an algebra extension and σ : K −→ K be a field homomorphism over F . Then σ is an isomorphism.

Proposition 8.2. Let K be the splitting field of {fi} over F . Given field extension L of K and field homomorphism σ : K −→ L over F , we have σ(K) = K. This is referred to as the stability of splitting fields.

Remark. Let K/F be a Galois extension with Galois group G = Gal(K/F ). G acts on {intermediate fields of K/F } by K1 7−→ σ(K1) where σ ∈ G and on {subgroups of G} by conjugation, i.e. H 7−→ σHσ−1.

Proposition 8.3. Let σ ∈ G.

0 −1 (i) If K1 is an intermediate field then (σ(K1)) = σK1σ (ii) For H ≤ G, σ(H0) = (σHσ−1)0

Theorem 8.1. Let K/F be a finite Galois extension. Let G be the Galois group Gal(K/L). If L is an intermediate field then L/F is Galois if and only if L0 = Gal(K/L) ⊆ G is a normal subgroup. In this case Gal(L/F ) = G/L0.

Note. So for a Galois extension, we have a bijection between Galois subfields and normal subgroups.

9 Separability

Proposition-Definition 9.1 (Inseparability). Let K/F be a field extension. Suppose char(F ) 6= p. We say α ∈ K is purely inseparable over F if there is some n ≥ 1 with αpn ∈ F . In this case, Gal(F (α)/F ) = 1.

Definition 9.1 (Derivative). Let F be a field. We can define derivatives as

n n df X X f 0(x) = = f jxj−1 ∈ F [x] if f(x) = f xj ∈ F [x] dx j j j=0 j=0

Remark. The following properties of the derivative holds for f, g ∈ F [x]:

(i) (f + g)0 = f 0 + g0

(ii) (fg)0 = fg0 + f 0g

(iii) α is a multiple root of f(x) ∈ F [x] if and only if 0 = f(α) = f 0(α)

6 Definition 9.2 (Separable Degree). Let K/F be a finite field extension. Its separable degree is [K : F ]S = number of field homomorphisms σ : K 7−→ F over F .

Proposition-Definition 9.2 (Separability). Let f(x) ∈ F [x] be irreducible. The following are equivalent. (i) In the splitting field L of f(x) over F , f(x) factorises into distinct linear factors

(ii) f 0(x) 6= 0

(iii) [F (α): F ] = [F (α): F ]S where α is a root of f(x) (iv) f(x) and α are separable over F

Definition 9.3 (Separability). A field extension K/F is separable if every α ∈ K is separable over F .

Theorem 9.1. Let K ⊇ L ⊇ F be a tower of finite field extensions. Then:

(i) [K : F ]S = [K : L]S[L : F ]S

(ii) [K : F ]S ≤ [K : F ] with equality occurring if and only if K/F is separable

Remark. In fact, suppose σi : K −→ F are distinct homomorphisms over L and τj : L −→ F distinct homomorphisms over F . We can then extend τj to τ j : F −→ F . It follows that τ jσi are all the distinct homomorphisms from K −→ F .

Corollary 9.1. Two results. (i) Consider finite field extensions K/L, L/F . Then K/F is separable if and only if K/L and L/F are separable

(ii) If a field K is generated over F by separable elements {αi} then K/F is separable

10 Criterion For Galois

Lemma 10.1. Let K/F be a Galois extension. Let L ⊆ K be a splitting field of some f(x) ∈ F [x] over F , then L/F is Galois.

Theorem 10.1. K/F is a Galois extension if and only if K is separable splitting field over F .

Proposition 10.1. The following are equivalent. (i) K/F is a normal extension, i.e. K is a splitting field over F for some family of polyno- mials

7 (ii) For any α ∈ K if fα(x) is its minimal polynomial over F then fα(x) factorises linearly over K

Theorem 10.2. Let K/F be an algebraic extension. For α ∈ K, let fα(x) ∈ F [x] be its minimal polynomial over F . Let L ⊆ K be the splitting field of {fα(x)}α∈K over F . So in particular L ⊇ K.

(i) If L1 is a subfield of K such that L1/F is normal and L1 ⊇ K then L1 ⊇ L (ii) If K/F is separable, so is L/F , i.e. L/F is Galois (iii) If K/F is finite, so is L/F and L is a splitting field of a single polynomial f(x) ∈ F [x] over F

Note. 10.2 (i) implies that L is the smallest splitting field containing K over F . For (iii), Qn the single polynomial f(x) is given by f(x) = i=1 fαi (x) where K = F (α1, α2, . . . , αn).

Definition 10.1 (Normal & Galois Closures). The field L in Theorem 10.2 is called the normal closure of K over F . If K/F is separable, then we also call L the Galois closure of K over F .

Remark. In a field of characteristic 0, where separability is guaranteed, you can always embed a finite extension in a finite Galois extension, i.e. Theorem 10.2 (iii).

11 Radical Extensions

Proposition 11.1. Let F be a field of characteristic p and n ≥ 2 be such that p - n. The ∗ ∼ group of roots of unity in F is µn = Z/nZ.

Lemma√ 11.1. Let p be a prime and F be a field with all the p roots of unity. for α ∈ F , let K = F ( p α). Then: (i) K/F is a splitting field for xp − α over F √ p ∼ ∼ (ii) If char(F ) 6= p and α∈ / F then K/F is Galois with Galois group Gal(K/F ) = µp = Z/pZ √ (iii) If char(F ) = p or p α ∈ F then Gal(K/F ) = 1

Lemma 11.2. Let p be a prime and F be a field. Let K be the splitting field of xp − 1 over F . Then Gal(K/F ) is abelian. In fact, K = F (ω) where ωp = 1.

Definition 11.1 (Solvability). Let G be a group. A normal chain of subgroups is a sequence of the form 1 = G0 E G1 E G2 E ... E Gn = G. We say G is solvable if there exists such a chain of normal subgroups with Gi+1/Gi of cyclic prime order or trivial.

Theorem 11.1. Consider tower of field extensions F = F ≤ F ≤ F ≤ ... ≤ F = K where √ 0 1 2 n p Fi+1 = Fi( i αi) for some prime pi and αi ∈ Fi. Suppose K/F is Galois and F contains all pith roots of unity. Then G = Gal(K/F ) is solvable.

8 12 Solvable Groups

Proposition 12.1. Let G be a finite group and N E G. Then G is solvable if and only if N and G/N are both solvable.

Proposition 12.2. A finite group G is solvable if and only if there is a normal chain of subgroups 1 = G0 E G1 E ... E Gn = G with all Gi+1/Gi abelian.

Definition 12.1 (Commutator Group). Let G be a group and g, h ∈ G. The commutator of g and h is [g, h] = g−1h−1gh. The commutator subgroup of G is the group generated by all [g, h] as g, h ranges over G. It is denoted by [G, G].

Proposition 12.3. Let G be a group.

(i) [G, G] E G (ii) G/[G, G] is abelian

(iii) Given any N E G with G/N abelian, we have N ≥ [G, G], i.e. G/[G, G] is the largest abelian quotient group

Definition 12.2 (Derived Series). Let G be a finite group. Its derived series is the normal (0) (1) (r+1) (r) (r) chain of subgroups G = G D G D ... where G = [G : G ].

Corollary 12.1. A finite group G is solvable if and only if in the derived series G(r) = 1 for some r < ∞.

Theorem 12.1. Let n ≥ 5.

(i) [An : An] = An

(ii) An and Sn are not solvable

13 Solvability By Radicals

Lemma 13.1. Let L be a Galois closure of a separable radical extension then L is radical.

Theorem 13.1. Let K/F be a field extension which embeds in separable radical extension L/F in the sense that we have a tower of field F ≤ K ≤ L. Then G = Gal(K/F ) is solvable.

Definition 13.1 (Solvability). Let F be a field and f(x) ∈ F [x] be irreducible. The Galois group of f(x) is Gal(K/F ) where K is the splitting field of f(x) over F . We say f(x) is solvable by radicals if K/F embeds in a separable radical extension L/F as in Theorem 13.1, i.e. Gal(K/F ) solvable.

Corollary 13.1. For p(x),F,K as in Definition 13.1. The Galois group of p(x) over F is Sn = Gal(K/F ). If n ≥ 5, p(x) is not solvable by radicals as Sn is not.

9 14 A Polynomial Not Solvable By Radicals

Lemma 14.1. Fix p a prime. Let σ, τ ∈ Sp be a p-cycle and a 2-cycle respectively. Then the subgroup generated by σ, τ is Sp

Proposition 14.1. Let f(x) ∈ Q[x] be irreducible of degree p. Suppose f(x) has exactly two non-real roots. Then the Galois group of f(x) is G = Sp. Thus f(x) is not solvable by radicals if p ≥ 5.

Theorem 14.1. Let K/F be a finite separable extension then K = F (α) for some α ∈ K. In particular, α is such that [F (α): F ] is maximum.

√ Theorem 14.2 (Fundamental Theorem Of Algebra). C = R[ −1] is algebraically closed.

Lemma 14.2. Let P be any 2-group. There is a chain of subgroups P = P0 EP1 EP2 E...E1 i with [P : Pi] = 2 .

15 Traces & Norms

Definition 15.1 (Trace & Norm). Let K/F be a finite field extension of degree n. Let α ∈ K and multiplication by α be mα : K −→ K an F-linear map. Represent mα by the n × n matrix over F , Mα. We define the trace of α to be trK/F (α) = tr(Mα) and the norm of α to be NK/F (α) = det(Mα).

Notation. For finite field extension L/F , let GL/F be the set of field homomorphisms L −→ L, so that |GL/F | = [L : F ]S.

Proposition 15.1. Let K/F be a finite separable extension. Then for α ∈ K, we have X Y tr(α) = σ(α) N(α) = σ(α)

σ∈GK/F σ∈GK/F

Theorem 15.1. Let F,K be fields. Consider distinct field homomorphisms X1,X2,...,Xn : F −→ K. These are linearly independent over F , i.e. there are a1, a2, . . . , an ∈ K with a1X1(α) + a2X2(α) + ... + anXn(α) = 0 for all α ∈ F then all the ai’s are zero.

Corollary 15.1. Let K/F be a finite separable extension. Then there exists an element α ∈ K with trK/F (α) = 1

10 16 Cyclic Extensions

Definition 16.1 (Cyclic Extension). A Galois extension is cyclic if its Galois group is either cyclic or abelian.

Theorem 16.1 (Hilbert’s Theorem 90). Let K/F be a cyclic extension of degree n with β Galois group G = hσi. Then α ∈ K satisfies N(α) = 1 if and only if α = σ(β) for some β ∈ K.

Theorem 16.2. Let K/F be a cyclic extension of degree n. Suppose charF - n and F contains all n nth roots of 1. Then K = F (β) where β has a minimal polynomial xn − b over β n F . In particular, we have β is such that ω = σ(β) , where ω = 1.

Theorem 16.3. Let K/F be a cyclic extension of degree n and with Galois group G = hσi. Then α ∈ K satisfies tr(α) = 0 if and only if α = β − σ(β).

Theorem 16.4. Let K/F be a Galois extension of prime degree p = char(F ). Then K = F (β) where β has minimum polynomial xp − x − b over F . In particular, we have β is such that 1 = β − σ(β).

Note. In Theorem 16.4, [K : F ] = p, prime. So |G| = p and G is cyclic of degree p. Thus K/F is also cyclic of degree p.

17 Solvable Extension

Definition 17.1 (Solvable Extension). Let K/F be a finite separable extension. We say K/F is solvable if its Galois closure has solvable Galois group.

Theorem 17.1. Let F be a field of characteristic 0. Then any solvable extension K/F embeds in a radical extension.

Lemma 17.1. Suppose K and F as in Theorem 17.1. Let n = |G| where G = Gal(K/F ). Let K1 and F1 be obtained from K and F respectively by adjoining all nth roots of unity. Then G1 = Gal(K1/F1) is also solvable.

Remark. Theorem 17.1 is false is char(F ) > 0 since there are cyclic√ extensions of degree p which cannot be constructed by adjoining pth roots (since xp −a = (x− p a)p is not separable). But if we only assume K/F finite and separable in Theorem 17.1, we do have another version of the theorem provided in our tower of field extensions, we allow not just adjoining pth roots but also roots of xp − x − a.

11 18 Finite Fields

Proposition 18.1. Let K be a finite field and p prime.

(i) char(K) = p > 0

(ii) |K| = p[K:Fp]

(iii) K is a subfield of Fp

Proposition-Definition 18.1 (Frobenius Norm). Let K be a field of characteristic prime p. The Frobenius homomorphism is the map φ : K −→ K; x 7−→ xp.

(i) This is a field homomorphism

(ii) If K is finite or is Fp then φ is an automorphism

Lemma 18.1. Let K be a field with pn elements, n a positive integer. Then any α ∈ K is a root of xpn − x = 0. Equivalently if φ is Frobenius homomorphism , φn(α) = α. In particular, φn is the identity on K.

Theorem 18.1. Let φ : Fp −→ Fp be the Frobenius homomorphism.

n hφ i pn (i) The fixed field Fpn = Fp is the splitting field of x − x over Fp

n (ii) |Fpn | = p

n (iii) If K is a subfield of Fp with p elements then K = Fpn

(iv) pn / p is a cyclic of degree n and has Galois group hφ| i = /n F F Fpn Z Z

(v) Fp/Fp is Galois

Note. Galois group of Fp/Fp is not hφi.

Corollary 18.1. Two useful facts.

(i) The subfields of Fpn are those of the form Fpd , d | n

n n (ii) Fp /Fpd is cyclic of degree d

Remark. The lattice of finite subfields of Fp is just the lattice of positive integers ordered by divisibility. i.e. if n, l are positive integers with lowest common multiple m, greatest common divisor d, the smallest subfield containing Fpn and Fpl is Fpm , while Fpn ∩ Fpl = Fpd .

12 19 Topological Groups

Definition 19.1 (Topological Group). A topological group is a group G endowed with a topology such that:

(i) Multiplication map µ : G × G −→ G is continuous

(ii) The inverse map η : G −→ G; g 7−→ g−1 is also continuous

Proposition 19.1. Let G be a topological group.

(i) For g ∈ G, left and right multiplication by g are bi-continuous

(ii) If U ⊆ G is open then gU is open for every g ∈ G

(iii) Let U ⊆ G be an open subgroup then U is closed

Q Proposition 19.2. Let {Gα}α∈A be a set of topological groups. Then G = α Gα endowed with the product topology is a topological group.

Notation. Consider data of:

(i) A partial ordered set (A, ≤)

(ii) For each α ∈ A, a group Gα

(iii) For each α, β ∈ A, with α ≤ β, a group homomorphism φαβ : Gα −→ Gβ such that for all α ≤ β ≤ γ, φαγ = φβγφαβ

φαγ Gα −−−−−→ Gγ

φαβ &%φβγ Gβ

Definition 19.2 (Direct & Inverse Systems). The above data are said to be:

(i) An inverse system of groups if for any β, γ ∈ A, there is some α ∈ A with α ≤ β, α ≤ γ

(ii) A direct system of groups if for any β, γ ∈ A, there is some δ ∈ A with δ ≥ β, δ ≥ γ

Proposition-Definition 19.1 (Inverse Limit). Let {Gα} be an inverse system of groups as Q Q in Definition 19.2. Let G = {(gα) ∈ α Gα : φαβ(gα) = gβ}. Then G is a subgroup of α Gα ←−− called the inverse limit of the inverse system {Gα} and denoted limα∈A Gα.

13 Proposition 19.3. Let (Gα, φαβ) be an inverse system of groups and G = lim←−− Gα ≤ Q α∈A α Gα. Let πα : G −→ Gα be the natural projection (gα) 7−→ gα. Suppose H is a group and for each α ∈ A, we have a group homomorphism ψα : H −→ Gα satisfying ψβ = φαβ ◦ ψα whenever α ≤ β. Then there is a unique group homomorphism ψ such that πα ◦ ψ = ψα for all α ∈ A. ψβ H −−−−−→ Gβ

ψα &%φαβ Gα

ψα H −−−−−→ Gα

ψ &%πα G

Remark. This is the universal property of inverse limits.

Definition 19.3 (Pro-Finite Group). A pro-finite group G is a group that is isomorphic to an inverse limit of an inverse system {Gα} of finite groups, i.e. all Gα are finite.

←−− Q Lemma 19.1. Let G = limα∈A Gα ≤ α Gα be a pro-finite group with Gα finite. Q (i) G ≤ α Gα is closed (ii) G is compact

(iii) The open subgroups are the closed subgroups of finite index

20 Infinite Galois Extensions

Lemma 20.1. Let K/F be a Galois extension with Galois group G. Let L be an intermediate field with L/F Galois too. Then the restriction map ρ : G = Gal(K/F ) −→ Gal(L/F ); 0 σ 7−→ σ|L is a well defined group homomorphism with kernel L = Gal(K/L). Also ρ is surjective.

Notation. K/F is a Galois field extension. We have the following setup for an inverse system of finite Galois groups.

1. A = set of subfields Kα of K such that Kα/F is finite Galois. Order A by inclusion. K/F is algebraic. This is a direct system. Why? Let L be the smallest field containing Kβ, Kγ ∈ A. Then L is finite being finitely generated by algebraic elements. And Galois closure of L is also finite over F .

2. Apply the Galois correspondence with Nα = Gal(K/Kα). If Nβ ⊆ Nα, there is a natural map φ : G/N −→ G/N ; gN 7−→ gN or Gal(K /F ) −→ Gal(K /F ); σ 7−→ σ . αβ β α β α β α |Kα Note Gal(Kα/F ) is finite, so this is an inverse system of finite groups.

14 ∼ ←−− Theorem 20.1. Let K/F be a Galois extension with Galois group G. Then G = limα∈A Gal(Kα/F ), where Kα is a subfield of K such that Kα is finite Galois. In particular, G is pro-finite.

Proposition-Definition 20.1 (Separable Closure). Let F be a field and F sep = set of ele- ments in F which are separable over F .

(i) F sep is a field

(ii) F sep/F is Galois. F sep is called the separable closure of F

Definition 20.1 (Absolute Galois Group). Let F be a field. the absolute Galois group of F is Gal(F sep/F ).

sep ˆ Note. Gal(Fp/Fp) = Gal(Fp /Fp) = Z as we have a correspondence between Fpm /Fp and Z/mZ.

21 Infinite Galois Groups

←−− Notation. Let G = limα∈A Gα be a pro-finite group with all Gα finite. Let πα : G −→ G be −1 the natural projection map. Note {1Gα } ≤ Gα is open. Thus Uα = ker(πα) = πα (1) is an open subgroup of G. It is often called the fundamental open neighbourhood of 1.

Proposition 21.1. With the above notation.

(i) The Uα form a basis of open neighbourhood of 1, i.e. if U is an open neighborhood of 1 then U ⊇ Uα for some α

(ii) For σ ∈ G, the {σGα} form a basis of open neighbourhoods (iii) G is Hausdorff, so in particular points are closed

Note. In topological spaces, {1} closed implies Hausdorff.

Lemma 21.1. Let K/F be a Galois extension with Galois group G. Let L be a an interme- diate field of K/F .

(i) Then L0 = Gal(K/L) ≤ G is topologically closed

(ii) The subspace topology on L0 is the same as that coming from its structure as a pro-finite group, i.e. from G

Theorem 21.1. Let K/F be a Galois extension. There are inverse bijections with Galois group G. {topologically closed subgroups H ≤ G} ←→ {intermediate fields L}; H 7−→ H0 = KH ; L 7−→ L0 = Gal(K/L).

15 22 Inseparability

Proposition 22.1. Let K/F be a field extension where char(F ) = p > 0.

(i) The following are equivalent

(a) α ∈ K is purely inseparable over F , i.e. αpn ∈ F for some n ≥ 0 (b) α is the only root of its minimal polynomial f(x) ∈ F [x]

(c) [F (α): F ]S = 1 (ii) In particular if α ∈ K is both separable and purely inseparable over F then α ∈ F

Proposition-Definition 22.1 (Maximal Separable Sub-Extension). Let F be a field of char- acteristic p > 0. Let K/F be a field extension. The intermediate field L = K ∩ F sep of all elements in K which are separable over F is such that K/L is purely inseparable. L is called the maximal separable sub-extension.

Proposition-Definition 22.2 (Maximal Purely Inseparable Sub-Extension). Let K/F be a field extension. The set of elements in K which are purely inseparable over F forms a field called the maximal purely inseparable sub-extension.

Theorem 22.1. Let F be a field of characteristic p > 0 and K/F is a normal extension with Galois group G = Gal(K/F ). Let Lsep = maximal separable sub-extension and Lpi = maximal purely inseparable sub-extension.

(i) K/KG is Galois with Galois group G

G (ii) Lpi = K

(iii) Lsep/F is Galois

(iv) Lsep ∩ Lpi = F

(v) The smallest subfield containing Lsep and Lpi is K

(vi) The restriction map φ : Gal(K/Lpi) −→ Gal(Lsep/F ) is a well defined group isomor- phism

K

separable %-inseprable Lpi Lsep

inseprable -% separable F

Corollary 22.1. A field F of characteristic p ≥ 0 is perfect if Frobenius homomorphism φ : F −→ F is surjective, i.e. automorphism. Also fields of characteristic 0 are said to be perfect too. Any finite extension K/F of a perfect field F is separable.

16 23 Duality

Definition 23.1 (Torsion). Let A be an abelian group and m be a positive integer. An element a ∈ A is m-torsion if ma = 0. We say A is m-torsion if every element in A is m-torsion. We say A is torsion if every element of A has finite order.

Proposition-Definition 23.1 (Set Of Group Homomorphisms). Let G be a group and A an abelian group. Let Hom(G, A) be the set of group homomorphism φ : G −→ A. This is an abelian group when endowed with addition (φ1 + φ2)(g) = φ1(g) + φ2(g), g ∈ G, φ1, φ2 ∈ Hom(G, A) and φ1(g) φ2(g) ∈ A.

Definition 23.2 (Dual). Let G be a torsion abelian group. Then the dual of G is G∧ = Hom(G, Q/Z).

Note. A∧ × B∧ = (A × B)∧ with bijection (φ, ψ) ←→ ρ where ρ(a, b) = φ(a) + ψ(b).

Definition 23.3 (Perfect Pairing). Let A, B be torsion abelian groups. A pairing between A, B is a function ψ : A × B −→ Q/Z such that: (i) ψ(a, ·), ψ(·, b) are group homomorphisms for all a ∈ A and b ∈ B (ii) if ψ(a, ·) = 0 =⇒ a = 0 and ψ(·, b) = 0 =⇒ b = 0 then ψ is said to be perfect

Proposition 23.1. Let A, B be finite abelian groups and ψ : A × B −→ Q/Z a perfect pairing between them. Then A ∼= B∧. In particular, the bijection is given by a ←→ ψ(a, ·).

Notation. Fix positive integer m. Let F be a field of characteristic not dividing m and such that F ≥ µm. Note that |µm| = m. Kummer theory classifies abelian extensions of F with m-torsion Galois groups. Here are some examples: 1. Let F ∗m = {αm : α ∈ F ∗}. F abelian =⇒ F ∗m ≤ F ∗ and F ∗/F ∗m is an m-torsion abelian group. √ √ ∗ ∗m ∗m m m ∗m 2. Let a ∈√F /F , say a = αF . We define a to be any mth root of a, i.e. αF . m ∗ Note a is well defined up to scalar√ multiple by some scalar β ∈ F . Why? Changing m ∗ m choices of mth root α changes√ α by an element of µm ⊆ F . Changing α to αβ where β ∈ F ∗, i.e. changes m a by β. √ ∗ ∗m m 3. Let J ≤ F /F . Define F ( J) to be the splitting field over√ F of the family of polynomials {xm − α : αF ∗m ∈ J}. This is generated over F by m a, a ∈ J.

√ Proposition 23.2. For J ≤ F ∗/F ∗m, F ( m J)/F is Galois.

√ ∗ ∗m m Proposition√ 23.3. Let J ≤ F /F . Let σ ∈ Gal(F ( J)/F ), a ∈ J. Then ψ(σ, a) = σ( m a) √ m a ∈ µm and is independent of the choice of the mth root of a.

√ Proposition 23.4. For J ≤ F ∗/F ∗m. G = Gal(F ( m J)/F ) is a m-torsion abelian.

17 24 Kummer Theory

Notation. Fix positive integer m and field of characteristic not dividing√ m such that F ∗ ∗m m contains all m mth root of unity. Let J ≤ F√/F and G = Gal(F ( J)/F ). We have a well σ( m a) √ ∼ ∼ 1 defined map ψ : G×J −→ µm;(σ, a) 7−→ m a . Note also that µm = Z/mZ = m Z/Z ⊆ Q/Z.

ψ Theorem√ 24.1. The map G × J −→ µm ,−→ Q/Z is a perfect pairing. If J is finite then Gal(F ( m J)/F ) ∼= J ∧ ∼= J.

Theorem 24.2. For positive integer m and field of characteristic not dividing m such that F contains all m mth root of unity, there is a bijection:

∗ ∗m (i) {subgroups J ≤ F /F } ←→ {abelian extensions of F with√ m-torsion Galois group} (considered up to isomorphisms of subfields of F ); J ←→ F ( m J)

(ii) finite groups correspond to finite extensions

18