# Part II Galois Theory

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1 Part II Galois Theory Theorems Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Groups, Rings and Modules is essential Field extensions, tower law, algebraic extensions; irreducible polynomials and relation with simple algebraic extensions. Finite multiplicative subgroups of a field are cyclic. Existence and uniqueness of splitting fields. [6] Existence and uniqueness of algebraic closure. [1] Separability. Theorem of primitive element. Trace and norm. [3] Normal and Galois extensions, automorphic groups. Fundamental theorem of Galois theory. [3] Galois theory of finite fields. Reduction mod p. [2] Cyclotomic polynomials, Kummer theory, cyclic extensions. Symmetric functions. Galois theory of cubics and quartics. [4] Solubility by radicals. Insolubility of general quintic equations and other classical problems. [3] Artin s theorem on the subfield fixed by a finite group of automorphisms. Polynomial invariants of a finite group; examples. [2] 1

2 Contents II Galois Theory (Theorems) Contents 0 Introduction 3 1 Solving equations 4 2 Field extensions Field extensions Ruler and compass constructions K-homomorphisms and the Galois Group Splitting fields Algebraic closures Separable extensions Normal extensions The fundamental theorem of Galois theory Finite fields Solutions to polynomial equations Cyclotomic extensions Kummer extensions Radical extensions Solubility of groups, extensions and polynomials Insolubility of general equations of degree 5 or more Computational techniques Reduction mod p Trace, norm and discriminant

3 0 Introduction II Galois Theory (Theorems) 0 Introduction 3

4 1 Solving equations II Galois Theory (Theorems) 1 Solving equations 4

5 2 Field extensions II Galois Theory (Theorems) 2 Field extensions 2.1 Field extensions Theorem (Tower Law). Let F/L/K be field extensions. Then [F : K] = [F : L][L : K] Lemma. Let L/K be a finite extension. Then L is algebraic over K. Proposition. Let L/K be a field extension, α L algebraic over K, and P α the minimal polynomial. Then P α is irreducible in K[t]. Theorem. Let L/K a field extension, α L algebraic. Then (i) K(α) is the image of the (ring) homomorphism φ : K[t] L defined by f f(α). (ii) [K(α) : K] = deg P α, where P α is the minimal polynomial of α over K. Corollary. Let L/K be a field extension, α L. Then α is algebraic over K if and only if K(α)/K is a finite extension. Theorem. Suppose that L/K is a field extension. (i) If α 1,, α n L are algebraic over K, then K(α 1,, α n )/K is a finite extension. (ii) If we have field extensions L/F/K and F/K is a finite extension, then F = K(α 1,, α n ) for some α 1,, α n L. Proposition (Eisenstein s criterion). Let f = a n t n + + a 1 t + a 0 Z[t]. Assume that there is some prime number p such that (i) p a i for all i < n. (ii) p a n (iii) p 2 a 0. Then f is irreducible in Q[t]. 2.2 Ruler and compass constructions Theorem. Let S R 2 be finite. Then (i) If R is 1-step constructible from S, then [Q(S {R}) : Q(S)] = 1 or 2. (ii) If T R 2 is finite, S T, and the points in T are constructible from S, Then [Q(S T ) : Q(S)] = 2 k for some k (where k can be 0). Corollary. It is impossible to double the cube. 5

6 2 Field extensions II Galois Theory (Theorems) 2.3 K-homomorphisms and the Galois Group 2.4 Splitting fields Lemma. Let L/K be a field extension, f K[t] irreducible, deg f > 0. Then there is a 1-to-1 correspondence Root f (L) Hom K (K[t]/ f, L). Corollary. Let L/K be a field extension, f K[t] irreducible, deg f > 0. Then In particular, if E = K[t]/ f, then Hom K (K[t]/ f, L) deg f. Aut K (E) = Root f (E) deg f = [E : K]. So E/K is a Galois extension iff Root f (E) = deg f. Theorem. Let K be a field, f K[t]. Then (i) There is a splitting field of f. (ii) The splitting field is unique (up to K-isomorphism). 2.5 Algebraic closures Lemma. If R is a commutative ring, then it has a maximal ideal. In particular, if I is an ideal of R, then there is a maximal ideal that contains I. Theorem (Existence of algebraic closure). Any field K has an algebraic closure. Theorem (Uniqueness of algebraic closure). Any field K has a unique algebraic closure up to K-isomorphism. 2.6 Separable extensions Lemma. Let K be a field, f, g K[t]. Then (i) (f + g) = f + g, (fg) = fg + f g. (ii) Assume f 0 and L is a splitting field of f. Then f has a repeated root in L if and only if f and f have a common (non-constant) irreducible factor in K[t] (if and only if f and f have a common root in L). Corollary. Let K be a field, f K[t] non-zero irreducible. Then (i) If char K = 0, then f is separable. (ii) If char K = p > 0, then f is not separable iff deg f > 0 and f K[t p ]. For example, t 2p + 3t p + 1 is not separable. Lemma. Let L/F/K be finite extensions, and E/K be a field extension. Then for all α L, we have Hom K (F (α), E) [F (α) : F ] Hom K (F, E). 6

7 2 Field extensions II Galois Theory (Theorems) Theorem. Let L/K and E/K be field extensions. Then (i) Hom K (L, E) [L : K]. In particular, Aut K (L) [L : K]. (ii) If equality holds in (i), then for any intermediate field K F L: (a) We also have Hom K (F, E) = [F : K]. (b) The map Hom K (L, E) Hom K (F, E) by restriction is surjective. Theorem. Let L/K be a finite field extension. Then the following are equivalent: (i) There is some extension E of K such that Hom K (L, E) = [L : K]. (ii) L/K is separable. (iii) L = K(α 1,, α n ) such that P αi, the minimal polynomial of α i over K, is separable for all i. (iv) L = K(α 1,, α n ) such that R αi, the minimal polynomial of α i over K(α 1,, α i 1 ) is separable for all i for all i. Lemma. Let L be a field, L \ {0} be the multiplicative group of L. If G is a finite subgroup of L, then G is cyclic. Theorem (Primitive element theorem). Assume L/K is a finite and separable extension. Then L/K is simple, i.e. there is some α L such that L = K(α). Corollary. Any finite extension L/K of field of characteristic 0 is simple, i.e. L = K(α) for some α L. Proposition. Let L/K be an extension of finite fields. Then the extension is separable. 2.7 Normal extensions Lemma. Let L/F/K be finite extensions, and K is the algebraic closure of K. Then any ψ Hom K (F, K) extends to some φ Hom K (L, K). Theorem. Let L/K be a finite extension. Then L/K is a normal extension if and only if L is the splitting field of some f K[t]. Theorem. Let L/K be a finite extension. Then the following are equivalent: (i) L/K is a Galois extension. (ii) L/K is separable and normal. (iii) L = K(α 1,, α n ) and P αi, the minimal polynomial of α i over K, is separable and splits over L for all i. Corollary. Let K be a field and f K[t] be a separable polynomial. Then the splitting field of f is Galois. 7

8 2 Field extensions II Galois Theory (Theorems) 2.8 The fundamental theorem of Galois theory Lemma (Artin s lemma). Let L/K be a field extension and H Aut K (L) a finite subgroup. Then L/L H is a Galois extension with Aut L H (L) = H. Theorem. Let L/K be a finite field extension. Then L/K is Galois if and only if L H = K, where H = Aut K (L). Theorem (Fundamental theorem of Galois theory). Assume L/K is a (finite) Galois extension. Then (i) There is a one-to-one correspondence H Aut K (L) intermediate fields K F L. This is given by the maps H L H Moreover, Aut K (L) : H = [L H : K]. and F Aut F (L) respectively. (ii) H Aut K (L) is normal (as a subgroup) if and only if L H /K is a normal extension if and only if L H /K is a Galois extension. (iii) If H Aut K (L), then the map Aut K (L) Aut K (L H ) by the restriction map is well-defined and surjective with kernel isomorphic to H, i.e. 2.9 Finite fields Aut K (L) H = Aut K (L H ). Lemma. Let X be a finite field with q = K element. Then (i) q = p d for some d N, where p = char K > 0. (ii) Let f = t q t. Then f(α) = 0 for all α K. Moreover, K is the splitting field of f over F p. Lemma. Let q = p d, q = p d, where d, d N. Then (i) There is a finite field K with exactly q elements, which is unique up to isomorphism. We write this as F q. (ii) We can embed F q F q iff d d. Theorem. Consider F q n/f q. Then Fr q is an element of order n as an element of Aut Fq (F q n). Theorem. The extension F q n/f q is Galois with Galois group Gal(F q n/f q ) = Aut Fq (F q n) = Z/nZ, generated by Fr q. 8

9 3 Solutions to polynomial equations II Galois Theory (Theorems) 3 Solutions to polynomial equations 3.1 Cyclotomic extensions Theorem. For each d N, there exists a dth cyclotomic monic polynomial φ d Z[t] satisfying: (i) For each n N, we have t n 1 = d n φ d. (ii) Assume char K = 0 or 0 < char K n. Then Root φn (L) = {nth primitive roots of unity}. Note that here we have an abuse of notation, since φ n is a polynomial in Z[t], not K[t], but we can just use the canonical map Z[t] K[t] mapping 1 to 1 and t to t. Theorem. Let K be a field with char K = 0 or 0 < char K n. Let L be the nth cyclotomic extension of K. Then L/K is a Galois extension, and there is an injective homomorphism θ : Gal(L/K) (Z/nZ). In addition, every irreducible factor of φ n (in K[t]) has degree [L : K]. Lemma. Under the notation and assumptions of the previous theorem, φ n is irreducible in K[t] if and only if θ is an isomorphism. Theorem. φ n is irreducible in Q[t]. In particular, it is also irreducible in Z[t]. Corollary. Let K = Q and L be the nth cyclotomic extension of Q. Then the injection θ : Gal(L/Q) (Z/nZ) is an isomorphism. 3.2 Kummer extensions Theorem. Let K be a field, λ K non-zero, n N, char K = 0 or 0 < char K n. Let L be the splitting field of t n λ. Then (i) L contains an nth primitive root of unity, say µ. (ii) L/K(µ) is a cyclic (and in particular Galois) extension with degree [L : K(µ)] n. (iii) [L : K(µ)] = n if and only if t n λ is irreducible in K(µ)[t]. Lemma. Assume L/K is a field extension. Then Hom K (L, L) is linearly independent. More concretely, let λ 1,, λ n L and φ 1,, φ n Hom K (L, L) distinct. Suppose for all α L, we have Then λ i = 0 for all i. λ 1 φ 1 (α) + + λ n φ n (α) = 0. Theorem. Let K be a field, n N, char K = 0 or 0 < char K n. Suppose K contains an nth primitive root of unity, and L/K is a cyclic extension of degree [L : K] = n. Then L/K is a Kummer extension. 9

10 3 Solutions to polynomial equations II Galois Theory (Theorems) 3.3 Radical extensions Lemma. Let L/K be a Galois extension, char K = 0, γ L and F the splitting field of t n γ over L. Then there exists a further extension E/F such that E/L is radical and E/K is Galois. Theorem. Suppose L/K is a radical extension and char K = 0. Then there is an extension E/L such that E/K is Galois and there is a sequence K = E 0 E 1 E, where E i E i+1 is cyclotomic or Kummer. 3.4 Solubility of groups, extensions and polynomials Lemma. Let G be a finite group. Then (i) If G is soluble, then any subgroup of G is soluble. (ii) If A G is a normal subgroup, then G is soluble if and only if A and G/A are both soluble. Lemma. Let L/K be a Galois extension. Then L/K is soluble if and only if Gal(L/K) is soluble. Theorem. Let K be a field with char K = 0, and L/K is a radical extension. Then L/K is a soluble extension. Corollary. Let K be a field with char K = 0, and f K[t]. If f can be solved by radicals, then Gal(L/K) is soluble, where L is the splitting field of f over K. Lemma. Let K be a field, f K[t] of degree n with no repeated roots. Let L be the splitting field of f over K. Then L/K is Galois and there exist an injective group homomorphism Gal(L/K) S n. Lemma. Let p be a prime, and σ S p have order p. Then σ is a p-cycle. Theorem. Let f Q[t] be irreducible and deg f = p prime. Let L C be the splitting field of f over Q. Let Root f (L) = {α 1, α 2,, α p 2, α p 1, α p }. Suppose that α 1, α 2,, α p 2 are all real numbers, but α p 1 and α p are not. In particular, α p 1 = ᾱ p. Then the homomorphism β : Gal(L/Q) S n is an isomorphism. 3.5 Insolubility of general equations of degree 5 or more Theorem (Symmetric rational function theorem). Let K be a field, L = K(x 1,, x n ). Let F the field fixed by the automorphisms that permute the x i. Then 10

11 3 Solutions to polynomial equations II Galois Theory (Theorems) (i) L is the splitting field of over F. f = t n e 1 t n ( 1) n e n (ii) F = L Sn L is a Galois group with Gal(L/F ) isomorphic to S n. (iii) F = K(e 1,, e n ). Theorem. Let K be a field with char K = 0. Then the general polynomial polynomial over K of degree n cannot be solved by radicals if n 5. Theorem. Let K be a field with char K = 0. If L/K is a soluble extension, then it is a radical extension. Corollary. Let K be a field with char K = 0 and h K[t]. Let L be the splitting of h over K. Then h can be solved by radicals if and only if Gal(L/K) is soluble. Corollary. Let K be a field with char K = 0. Let f K[t] have deg f 4. Then f can be solved by radicals. 11

12 4 Computational techniques II Galois Theory (Theorems) 4 Computational techniques 4.1 Reduction mod p Theorem. G = {λ S n : λ preserves the irreducible factor corresponding to G}. ( ) Theorem. Let f Z[t] be monic with no repeated roots. Let E be the splitting field of f over Q, and take f F p [t] be the obvious polynomial obtained by reducing the coefficients of f mod p. We also assume this has no repeated roots, and let Ē be the splitting field of f. Then there is an injective homomorphism Ḡ = Gal(Ē/F p) G = Gal(E/Q). Moreover, if f factors as a product of irreducibles of length n 1, n 2,, n r, then Gal(f) contains an element of cycle type (n 1,, n r ). 4.2 Trace, norm and discriminant Lemma. Let L/F/K be finite field extensions. Then tr L/K = tr F/K tr L/F, N L/K = N F/K N L/F. Lemma. Let F/K be a field extension, and V an F -vector space. Let T : V V be an F -linear map. Then it is in particular a K-linear map. Then det K T = N F/K (det F T ), tr K T = tr F/K (tr F T ). Corollary. Let L/K be a finite field extension, and α L. Let r = [L : K(α)] and let P α be the minimal polynomial of α over K, say with a i K. Then and P α = t n + a n 1 t n a 0. tr L/K (α) = ra n 1 N L/K (α) = ( 1) nr a r 0. Theorem. Let L/K be a finite but not separable extension. Then tr L/K (α) = 0 for all α L. Theorem. Let L/K be a finite separable extension. Pick a further extension E/L such that E/K is normal and Hom K (L, E) = [L : K]. Write Hom K (L, E) = {ϕ 1,, ϕ n }. Then for all α L. n n tr L/K (α) = ϕ i (α), N L/K (α) = ϕ i (α) i=1 i=1 12

13 4 Computational techniques II Galois Theory (Theorems) Corollary. Let L/K be a finite separable extension. Then there is some α L such that tr L/K (α) 0. Theorem. Let K be a field and f K[t], L is the splitting field of f over K. Suppose D f 0 and char K 2. Then (i) D f K. (ii) Let G = Gal(L/K), and θ : G S n be the embedding given by the permutation of the roots. Then im θ A n if and only if f K (if and only if D f is a square in K). Theorem. Let K be a field, and f K[t] be an n-degree monic irreducible polynomial with no repeated roots. Let L be the splitting field of f over K, and let α Root F (L). Then D f = ( 1) n(n 1)/2 N K(α)/K (f (α)). 13

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