AN INTRODUCTION TO GALOIS THEORY STEVEN DALE CUTKOSKY In these notes we consider the problem of constructing the roots of a polynomial. Sup- pose that F is a subfield of the complex numbers, and f(x) is a polynomial over F. We wish to give a rational formula for constructing the roots of f(x). The quadratic formula is an example of such a formula. Let f(x) = ax2+bx+c, where a 6= 0. We make the substitution y = x+ b to obtain 2a the equation b2 c y2 = − , 4a2 a which yields a formula for the two roots of f(x), √ −b± b2−4ac x = . 2a This formula was known in ancient times, while formulas for solving cubic and quadric equations where discovered in the Renaissance. Suppose that f(x) = ax3 +bx2 +cx+d, with a 6= 0. We first make the substitution y = x+ b to obtain the equation 3a y3+py+q = 0 (with p = c − b2 , q = d − bc + 2b3 ). Now make the substitution y = z− p to obtain a 3a2 a 3a2 27a3 3z p3 z3− +q = 0. 27z3 We multiply the equation by z3 and use the quadratic formula to obtain r q q2 p3 z3 = − ± + . 2 4 27 Taking cube roots, we obtain six solutions for z. Substituting into y = z − p we obtain 3z 3 distinct solutions. Next, we solve the quartic equation. Suppose that f(x) = ax4+bx3+cx2+dx+e with a 6= 0. substitute z = x+ b to obtain the equation 4a z4+pz2+qz+r = 0. For all u ∈ C, this expression is equal to u2 u2 (1) z4+z2u+ −z2u− +pz2+qz+r = 0, 4 4 or u u2 (z2+ )2−[(u−p)z2−qz+( −r)] = 0. 2 4 Date: November 16, 2009. 1 LetP = z2+u. Thequadraticpolynomial(u−p)z2−qz+(u2−r)isoftheformQ2, where 2 4 Q is a linear polynomial in z, precisely when the discriminant of (u−p)z2−qz+(u2 −r) 4 is zero. This occurs for values of u such that u2 q2 = 4(u−p)( −r). 4 Solving this cubic equation for u, we set u to be a root of this equation, so that (1) has the form 0 = P2−Q2 = (P +Q)(P −Q), or u u (2) (z2+ +L)(z2+ −L) = 0 2 2 where L is a linear function of z. Equation (2) is a product of two quadratic equations, each of which can be solved for z using the quadratic formula. We have thus obtained the solutions to our quartic equation f(x). At this point, one may wonder if, with sufficient cleverness, it may be possible to solve fifth and higher degree equations. This is actually not the case, as was discovered only in the nineteenth century, by Abel, Ruffini and Galois. The remainder of these notes will be devoted to proving the insolvability of higher degree equations. All rings R will be assumed to have a multiplicative identity 1 . We will denote 1 R R by 1 when there is no danger of confusion. All ring homomorphisms Φ : R → S will be required to satisfy Φ(1 ) = 1 . In particular, if R is a field, then Φ is 1-1. R S Suppose that R,S are commutative rings (with identity), and U is a subset of S. We define the subring of S generated by R and U by Xr1 Xrn R[U] = { ··· a ui1···uin | n ∈ N,u ,...,u ∈ U,r ,...,r ∈ N and a ∈ R}. i1···in 1 n 1 n 1 n i1···in i1=0 rn=0 If U = {u}, r X R[u] = { a ui | r ∈ N and a ∈ R}. i i i=0 Suppose that K is a subfield of a field L, and U is a subset of L. We define the subfield of L generated by K and U to be f K(U) = { | f,g ∈ K[U] and g 6= 0}. g Thepolynomialringintheindeterminates{x ,...,x }isaringR[x ,...,x ]generated 1 n 1 n byRandtheset{x ,...,x }whichhasthepropertythatforapolynomialwithcoefficients 1 n a ∈ R, i1···in Xr1 Xrn f(x ,...,x ) = ··· a xi1···xin = 0 1 n i1···in 1 n i1=0 rn=0 if and only if all coefficients a = 0. i1···in The polynomial ring R[x ,...,x ] has the following universal property. 1 n Theorem 0.1. Suppose that Φ : R → S is a ring homomorphism (of commutative rings with identity), and u ,...,u ∈ S. Then there is a unique ring homomorphism Φ : 1 n R[x ,...,x ] → S such that Φ(r) = r for r ∈ R and Φ(x ) = u for 1 ≤ i ≤ n. 1 n i i 2 SupposethatF isasubfieldofafieldK, andu ∈ K. uiscalledalgebraicoverF ifthere existapositiveintegernanda ,a ,...,a ∈ F notallzerosuchthata +a +···+a un = 0 1 n 0 u n 0. u is called transcendental over F if u is not algebraic over F. By the universal property of polynomial rings, there is a surjective ring homomorphism Ψ : F[x] → F[u] such that Ψ(f(x)) = f(u) for f(x) ∈ F[x]. Let I be the kernel of Ψ. Since F[x] is a PID, I = (f) = {fg | g ∈ F[x]} for some f ∈ F[x]. By the first isomorphism theorem, Ψ : F[x]/I → F[u] defined by Ψ(g +I) = Ψ(g) is a ring isomorphism. Since F[u] is a domain, I is a prime ideal in F[x] and thus either I is the zero ideal, or f(x) is irreducible in F[x]. There is a relation a +a u+···+a un = 0 in F[u] with n a nonnegative integer and 0 1 n a ∈ F for all i if and only if the polynomial g(x) = a +a x+···+a xn is in the kernel i 0 1 n I of Ψ. Thus u is transcendental over F if and only if the Kernel I of Ψ is the zero ideal. If u is transcendental over F, then F[u] is (isomorphic to) a polynomial ring, so F[u] is not a field. The field F(u) generated by F and u is the rational function field f(x) F(u) = { | f,g ∈ F[x] and g 6= 0}. g(x) The statement that u is transcendental over F is equivalent to the statement that the set {ui | i ∈ N} is a basis of F[u]. Suppose that u is algebraic over F, so that the kernel I of Ψ is generated by an ir- reducible polynomial f(x). Since the units in F[x] are the nonzero elements of F, there is a unique monic polynomial f(x) which generates I. A polynomial f(x) is monic if its leading coefficient is 1. That is, f(x) has an expression f(x) = a +a x+···+a xn−1+xn 0 1 n−1 for some n ≥ 1. This unique monic polynomial f(x) which generates the kernel I of Ψ is called the minimal polynomial of u. We say that u is a root of a polynomial g(x) ∈ F[x] if g(u) = 0 (or equivalently g ∈ I). Theorem 0.2. Suppose that F is a field and u is algebraic over F. Let f(x) be the minimal polynomial of u over F. Then 1. f(x) is the monic polynomial in F[x] of smallest degree with the property that f(u) = 0. 2. f is irreducible in F[x]. 3. u is a root of g(x) ∈ F[x] if and only if f divides g. The following criteria is useful for finding the minimal polynomial. Corollary 0.3. Suppose that F is a field and u is algebraic over F. A polynomial f(x) ∈ F[x] is the minimal polynomial of u over F if and only if 1. u is a root of f. 2. f is monic and irreducible in F[x] Lemma 0.4. Suppose that R is a domain which contains a field F such that R is a finite dimensional vector space over F. Then R is a field. 3 Proof. Suppose a ∈ R is a nonzero element. Define T : R → R by T(b) = ab for b ∈ R. The kernel of T is {0} since T is a domain. Since R is a finite dimensional vector space, T is onto by the rank nullity theorem. Thus there exists b ∈ R such that ab = 1. (cid:3) Lemma 0.5. SupposethatuisalgebraicoverF withminimalpolynomialf(x) ∈ F[x]. Let n = deg(f). Then F[u] is an n-dimensional vector space over F. Further, {1,u,...,un−1} is an F-basis of F. Thus F[u] is a field. Proof. Suppose that β ∈ F[u]. Then β = g(u) for some polynomial g(x) ∈ F[x]. By Euclidean division, g(x) = q(x)f(x) + r(x) where q,r ∈ F[x] and deg(r) < n. β = g(u) = q(u)f(u) + r(u) = r(u). Thus {1,u,u2,...,un−1} span F[u] as a vector space over F. Suppose that there is a relation a +a u+a u2 +···+a un−1 = 0 for some 0 1 2 n−1 a ,a ,...,a ∈ F. Letg(x) = a +a x+···+a xn−1 ∈ F[x]. Sinceg(u) = 0, wehave 0 1 n−1 0 1 n−1 that f divides g in F[x]. If g 6= 0 we must then have that n−1 ≥ deg(g) ≥ deg(f) = n, a contradiction. Thus we must have that g = 0, and we see that {1,u,...,un−1} are linearly independent over F and are thus a basis of F[u]. F[u] is a field by Lemma 0.4. (cid:3) Definition 0.6. Suppose that F is a subfield of a field K, so that K is a vector space over F. Denote the dimension of K as a vector space over F by [K : F]. Thus u is algebraic over F if and only if [F[u] : F] < ∞. Ausefulcriterionforirreducibilityofpolynomialsofsmalldegreeisthefollowinglemma. Lemma 0.7. Suppose that F is a field and f(x) ∈ F[x] is a nonzero polynomial of degree 2 or 3. Then f(x) is irreducible if and only if f has no root in F. Example 0.8. The above criterion is not valid for polynomials of degree ≥ 4. A simple example is f(x) = (x2+1)2 ∈ Q[x] which has no root in Q, but is not irreducible. Lemma 0.9. Suppose that f(x) = a +a x+···+a xn with a ∈ Z and a 6= 0. Suppose 0 1 n i n that γ ∈ Q is a root of f(x). Write γ = α where α,β are relatively prime integers. Then β α divides a and β divides a . 0 n √ Example 0.10. f(x) = x2+1 is the minimal polynomial of −1 over Q. [Q[i] : Q] = 2. The n-th roots of unity in a field F is the set {ξ ∈ F | ξn = 1}. In C, the n-th roots of √ m2π −1 unity is the set {e n | 0 ≤ m ≤ n−1}. √ 2π −1 Example 0.11. Suppose that p is a prime number. Let ξ = e p ∈ C. ξp = 1 so that ξ is a root of the polynomial xp−1 ∈ Q[x]. xp−1 factors as the product xp−1 = (xp−1+xp−2+···+x+1)(x−1) in Q[x]. Since ξ −1 6= 0, ξ is a root of the cyclotomic polynomial f(x) = xp−1 +xp−2 + ···+x+1. Since p is a prime, f(x) is irreducible in Q[x] by Eisenstein’s criterion. Thus f(x) is the minimal polynomial of ξ over Q. Q[ξ] ∼= Q[x]/(f(x)) and [Q[ξ] : Q] = p−1. √ √ ReturningtoExample0.10, observethat( −1)4 = 1but[Q[ −1] : Q] = 2, not3. This is because the cyclotomic polynomial x3+x2+x+1 = (x+1)(x2+1) is not irreducible. Lemma 0.12. Suppose that K ⊂ L ⊂ M are fields. Then [M : K] = [M : L][L : K]. 4 Lemma 0.12 is proven by showing that if {u } is a basis of L over K and {v } is a basis i j of M over L, then {u v } is a basis of M over K. i j Definition 0.13. Suppose that F is a field and f(x) ∈ F[x] is a nonzero polynomial. An extension field E of F is a splitting field of f(x) over F if the following two conditions hold: 1. f(x) factors into a product of linear factors in E[x]. That is, there exist distinct elements u ,...,u ∈ E, a nonzero element c ∈ F and positive integers e ,...,e 1 k 1 k such that (3) f(x) = c(x−u )e1(x−u )e2···(x−u )ek. 1 2 k 2. E = F[u ,...,u ] (so that E is generated by the roots of f(x) and F). 1 k We say that f has multiple roots if some e > 1 in (3). i Theorem 0.14. Suppose that F is a field and f(x) is an irreducible polynomial in F[x]. Then there exists an extension field K = F[α] of F such that f(α) = 0. Proof. Set K = F[x]/(f(x)). Let α = x+(f(x)). We have f(α) = f(x)+(f(x)) = 0. (cid:3) Theorem 0.15. Suppose that F is a field, and f(x) ∈ F[x] is a nonzero polynomial. Then there exists a splitting field E of f(x) over F. Proof. Weprovethetheorembyinductiononthedegreeoff(x). Letp(x)beanirreducible factor of f(x) in F[x]. By Theorem 0.14 there exists an extension field K = F[α] of F such that α is a root of p(x). Thus f(x) is a product of the linear polynomial (x−α) and a polynomial g(x) ∈ K[x]. By induction, there exists a splitting field E of g(x) over K. Thus E is a splitting field of f(x) over F. (cid:3) Example 0.16. Suppose that Suppose that f(x) = ax2 + bx + c ∈ Q[x] is irreducible. Then b2−4ac 6= 0, so that f has two distinct roots √ √ −b+ b2−4ac −b− b2−4ac u = and u = 1 2 2a 2a √ (where b2−4ac is a choice of square root). K = Q[u ,u ] is a splitting field of f(x) 1 2 over Q. We have K = Q[u ] since u = −u − b ∈ Q[u ]. [K : Q] = 2 since Q[u ] ∼= 1 2 1 a 1 1 Q[x]/(f(x)). √ Example 0.17. Let f(x) = x3 −2 ∈ Q[x]. Let u = 3 2 be the (unique) real cube root of √ 2. Let ω = e2π3−1. f(x) = (x−u)(x−ωu)(x−ω2u) in C[x]. Thus K = Q[u,ωu,ω2u] = Q[u,ω] is a splitting field of f(x) over Q. Observe that Q[u] is not equal to K, since Q[u] is contained in R, which does not contain ω. As f(x) has degree 3 and has no root in Q, f(x) is irreducible in Q[x] and is the minimal polynomial of u over Q. Thus Q[u] ∼= Q[x]/(x3 −2), so [Q[u] : Q] = 3. Let L = Q[u]. x2 +x+1 = (x−ω)(x−ω2). x2 +x+1 is irreducible in L[x] since x2 +x+1 has degree 2 and x2 +x+1 does not have a root in L. Thus K = L[ω] ∼= L[x]/(x2 + x + 1), and [K : L] = 2. We have [K : Q] = [K : L][L : Q] = 6. Example 0.18. Let f(x) = x3+x+1 ∈ Z [x]. f(0) = f(1) = 1 so f has no roots in Z . 2 2 Since f has degree 3, f(x) is irreducible in Z [x]. Let K = F[x]/(f(x)), an extension field 2 5 of Z . Set α = x+(f(x)). K = Z [α]. α is a root of f(x) since α3+α+1 = 0. We have 2 2 a factorization x3+x+1 = (x+α)(x+α2)(x+α+α2). Thus K is a splitting field of f(x) over Z . [K : Z ] = 3. K consists of the 8 elements 2 2 K = {s+tα+uα2 | s,t,u ∈ Z }. 2 Definition 0.19. A field F is algebraically closed if f(x) ∈ F[x] and deg(f) > 0 implies f(x) has a root in F. By Euclidian division, a field F is algebraically closed if and only if every polynomial f(x) ∈ F[x] factors into a product of linear factors in F[x]. Definition 0.20. A field K is an algebraic closure of a field F if K is algebraic over F and K is algebraically closed. The algebraic closure of a field is uniquely determined up to isomorphism. Theorem 0.21. Every field has an algebraic closure. The idea of the proof of Theorem 0.21 is to successively apply Theorem 0.15 to all poly- nomials in F[x]. This requires some care, as F could have a large cardinality. Bythe fundamentaltheoremofalgebra, Cisalgebraicallyclosed. AsCisalgebraic over R, C is the algebraic closure of R. Define Q = {u ∈ C | u is algebraic over Q}. Q is an algebraic closure of Q. As there are transcendental numbers such as π and e, Q is strictly smaller than C. Since Q is countable, the polynomials Q[x] are countable. Since every polynomial of a fixeddegreedhasatmostdroots,weconcludethatQiscountable. SinceRisuncountable, C is uncountable. Thus C is vastly larger than Q. Definition 0.22. Suppose that F is a subfield of a field K. t ,...,t are algebraically 1 r independent over F if f(t ,...,t ) 6= 0 for every nonzero polynomial f(x ,...,x ) in the 1 r 1 r polynomial ring F[x ,...,x ]. 1 r It follows from the definition that the surjective ring homomorphism F[x ,...,x ] → 1 r F[t ,...,t ] defined by f(x ,...,x ) 7→ f(t ,...,t ) is an isomorphism if and only if 1 r 1 r 1 r t ,...t are algebraically independent over F. 1 r Lemma 0.23. Suppose that r is a positive integer. Let L be a subfield of C which is algebraic over Q. Then there exist t ,...,t ∈ C which are algebraically independent over 1 r L. Proof. Suppose that there exists a number r such that there do not exist t ,...,t ∈ C 1 r which are algebraically independent over L. Let r be the smallest number with this property. Set s = r−1. Then there exist elements t ,...,t in C such that t ,...,t are 1 s 1 s algebraically independent over L and C is algebraic over the field K = L(t ,...,t ). Thus 1 s 6 C is an algebraic closure of K. K is countable since L is countable (as L ⊂ Q). Thus the algebraic closure C of K is countable (by the argument we used to show that Q is countable). This is a contradiction, since C is uncountable. (cid:3) Definition 0.24. Suppose that F is a subfield of a field K. The F-automorphisms of K is the group Aut K = { field automorphisms σ : K → K | σ(a) = a for a ∈ F}. F Suppose that η : F → K is a homomorphism of fields. We have an induced homomor- phism of polynomial rings η : F[x] → K[x], defined by η(f(x)) = η(a )+η(a )x+···+η(a )xn 0 1 n for f(x) = a +a x +···+a xn ∈ F[x], 0 1 1 n with a ,...,a ∈ F. Denote η(f(x)) by fη(x). We have (fg)η = fηgη for f,g ∈ F[x]. 0 n Theorem 0.25. Suppose that η : F → F∗ is an isomorphism of a field F onto a field F∗. Let K be an extension field of F and K∗ be an extension field of F∗. Suppose that b ∈ K is algebraic over F with minimal polynomial g(x). Then η can be extended to a homomorphism ζ : F[b] → K∗ if and only if gη(x) has a root in K∗. If gη(x) has a root in K∗, then the number of such extensions is equal to the number of distinct roots of gη(x) in E∗. Proof. Suppose that an extension ζ : F[b] → K∗ exists. Write g(x) = a +a x+a x2+···+a xn−1+xn 0 1 2 n−1 with a ∈ F. Then i 0 = ζ(0) = ζ(g(b)) = ζ(a +a b+a b2+···+a bn−1+bn) 0 1 2 n−1 = ζ(a )+ζ(a )ζ(b)+ζ(a )ζ(b)2+···+ζ(a )ζ(b)n−1+ζ(b)n 0 1 2 n−1 = η(a )+η(a )ζ(b)+η(a )ζ(b)2+···+η(a )ζ(b)n−1+ζ(b)n 0 1 2 n−1 = gη(ζ(b)). Thus ζ(b) is a root of gη in K∗. We further have that ζ is completely determined by ζ(b). Conversely, suppose that α is a root of gη(x) in K∗. gη(x) is the minimal polynomial of α over F∗ since gη(x) is irreducible in F∗[x] (as η is an isomorphism). Thus F∗[α] ∼= F∗[x]/(gη(x)). The isomorphism η : F[x] → F∗[x] takes the ideal (g(x)) to the ideal (gη(x)). Thus we have an induced isomorphism F[x]/(g(x) ∼= F∗[x]/(gη(x)). Since F[b] ∼= ∼ F[x]/(g(x)), we may compose these isomorphisms to obtain an isomorphism ζ : F[b] = F∗[α] which extends η, and such that ζ(b) = α. (cid:3) Corollary 0.26. Suppose that F is a field, and f(x) is an irreducible polynomial in F[x]. Let K = F[x]/(f(x)). Then |Aut K| is the number of distinct roots of f(x) in K. F Theorem 0.27. (Extension Theorem) Suppose that η : F → F∗ is an isomorphism of a field F onto a field F∗. Suppose that f(x) ∈ F[x] is a nonzero polynomial, and let fη(x) be the corresponding polynomial in F∗. Let E be a splitting field of f(x) over F and E∗ be a splitting field of fη(x) over F∗. Then η can be extended to an isomorphism of E onto E∗. Moreover, the number of such extensions is less than or equal to [E : F] and it is precisely [E : F] if fη(x) has no multiple roots in E∗. 7 Proof. We prove the theorem by induction on [E : F]. Let p(x) be an irreducible factor of f(x) in F[x] of degree d ≥ 1. Then pη is an irreducible factor of fη in F∗[x]. pη has e ≤ d = [F[u] : F] distinct roots v ,...,v in E∗. Let u be a root of p in E. 1 e By Theorem 0.25, there are e distinct extensions of η to a homomorphism from F[u] to E∗, η : F[u] → F∗[v ], where η (u) = v . These are all of the extensions of η to a i i i i homomorphism η0 : F[u] → E∗ since η0(u) must be a root of pη in E∗. By induction (replacing F with F[u] and F∗ with F∗[v ]), for 1 ≤ i ≤ e, there are less than or equal to i [E : F[u]] extensions of η : F[u] → E∗ to a homomorphism E → E∗. Hence there are less i than or equal to [E : F[u]][F[u] : F] = [E : F] extensions of η to E, with equality if fη(x) has no multiple roots in E∗. Such an extension η0 of η to E takes a basis of E as a vector space over F to a basis of the image of η0 as a vector space over F∗. Thus [E : F] ≤ [E∗ : F∗]. Applying the above proof to η−1 : F∗ → F, we see that [E∗ : F∗] ≤ [E : F], so that [E : F] = [E∗ : F∗]. Thus all extensions of η to a homomorphism η0 from E to E∗ are isomorphisms. (cid:3) Remark 0.28. We will see in Corollary 0.37, that if F is a field of characteristic zero, thenanysplittingfieldE overF ofapolynomialinF[x]isthesplittingfieldofapolynomial with no multiple roots. Theorem 0.29. Suppose that K is a splitting field of a nonzero polynomial f(x) ∈ F[x] of degree n. Then Aut K is isomorphic to a subgroup of S and thus |Aut K| ≤ n!. F n F Proof. Let u ,...,u be the distinct roots of f(x) in K, so that K = F[u ,...,u ]. Write 1 k 1 k f(x) = a +a x+a x2+···+a xn with a ∈ F and a 6= 0. Suppose that σ ∈ Aut K. 0 1 2 n i n F Then for 1 ≤ i ≤ k, 0 = σ(0) = σ(f(u )) = σ(a +a u +a u2+···+a un) i 0 1 i 2 i n i = σ(a )+σ(a )σ(u )+σ(a )σ(u )2+···+σ(a )σ(u )n 0 1 i 2 i n i = a +a σ(u )+a σ(u )2+···+a σ(u )n 0 1 i 2 i n i = f(σ(u )). i Thus an element σ ∈ Aut (K) permutes the roots of f(x). Further, since K is generated F by {u ,...,u } and F, σ fixes all of these roots if and only if σ is the identity map. Thus 1 k the map Λ : Aut K → Sym({u ,...,u }) defined by restriction to the set {u ,...,u } F 1 k 1 k is a 1-1 group homomorphism. If k ≤ n, we can include {u ,...,u } into a set with n 1 k elements to get an inclusion of groups Sym({u ,...,u }) into S . (cid:3) 1 k n Example 0.30. Recall from Example 0.17 that K = Q[u,ω] is a splitting field of x3 −2 √ √ over Q, where u = 3 2 is the real cube root of 2 and ω = e2π3−1. Let L = Q[u]. [L : Q] = 3 but AutQL = {id} by Corollary 0.26 since u is the only root of x3−2 in L. We will determine AutQK by computing its action on the set of roots {u,ωu,ωu2} of x3 −2. Recall from Example 0.17 that K = L[ω] ∼= L[x]/(x2 +x+1). Since ω,ω2 ∈ K are the two roots of x2 + x + 1, it follows from Theorem 0.25 and Corollary 0.26 that Aut K ∼= Z with generator σ which satisfies σ(ω) = ω2 and σ(t) = t for t ∈ L. Thus L 2 σ(u) = u,σ(uω) = uω2,σ(uω2) = uω. Let M = Q[ω]. [M[u] : M] = [K : M] = [K : Q]/[M : Q] = 3. Thus the minimal polynomial of u over M has degree 3. Since u is a root of the degree 3 monic polynomial x3 −2 ∈ M[x], x3 −2 is the minimal polynomial of u over M. Thus 8 K = M[u] ∼= M[x]/(x3 −2). By Corollary 0.26, Aut K ∼= Z since x3 −2 has 3 roots M 3 in K. By Theorem 0.25, a generator of Aut K is the M-automorphism τ which satisfies M τ(u) = ωu. Thus τ(u) = ωu,τ(uω) = uω2,τ(uω2) = u. We have inclusions of groups AutLK ⊂ AutQK and AutMK ⊂ AutQK. Thus AutQK contains an element of order two and an element of order 3. Since AutQK is a subgroup ∼ of S3 by Theorem 0.29, we have that AutQK = S3 is generated by σ and τ. Definition 0.31. Suppose that F is a field. A nonzero polynomial f(x) ∈ F[x] of degree n is separable over F if it has n distinct roots in a splitting field N of f over F. If f(x) is not separable over F, it is called inseparable. Definition 0.32. Suppose that F is a field, and f(x) ∈ F[x] is a polynomial. Write f(x) = a +a x+···+a xn with a ∈ F. The formal derivative of f(x) is 0 1 n i f0(x) = a +2a x+3a x2+···+na xn−1. 1 2 3 n Lemma 0.33. Suppose that f,g ∈ F[x]. Then 1. (f+g)’=f’+g’. 2. (fg)0 = fg0+f0g. 3. Suppose that m ∈ N. Then (fm)0 = mfm−1f0. Suppose that F is a subfield of a field K, and f(x),g(x) ∈ F[x] are polynomials. Recall that a greatest common divisor of f(x) and g(x) in F[x] is a greatest common divisor of f(x) and g(x) in K[x]. To prove this, observe that a greatest common divisor of f(x) and g(x) in F[x] is a generator d of the principal ideal generated by f and g in F[x]. Thus d is also a generator of the ideal generated by f and g in K[x], and thus d is a greatest common divisor of f and g in K[x] also. Theorem 0.34. Suppose that F is a field and f(x) ∈ F[x] is a nonzero polynomial. Then f(x) is separable over F if and only if 1 is a greatest common divisor of f and f0. Proof. Let N be a splitting field of f over F. Let u ,...,u be the distinct roots of f(x) 1 k in N. Then there exist positive integers e ,...,e and a nonzero c ∈ F such that 1 k f(x) = c(x−u )e1(x−u )e2···(x−u )ek. 1 2 k We have f0(x) = c(cid:0)e (x−u )e1−1(x−u )e2···(x−u )ek +(x−u )e1[(x−u )e2···(x−u )ek]0(cid:1). 1 1 2 k 1 2 k Thus (x−u ) divides f0(x) in N[x] if and only if e > 1. The same argument applied to 1 1 other factors of f(x) shows that x−u divides f0(x) in N[x] if and only if e > 1. Since i i the only irreducible factors of f(x) in N[x] are x − u , 1 ≤ i ≤ k, we have that 1 is a i greatest common divisor of f and f0 if and only if e = 1 for 1 ≤ i ≤ k. (cid:3) i Corollary 0.35. Suppose that F is a field and f(x) ∈ F[x] is an irreducible polynomial. Then f(x) is separable unless its formal derivative is zero. Proof. Suppose that f(x) is not separable. Since 1 is not a greatest common divisor of f and f0, and f(x) is irreducible, we have that f(x) divides f0(x). Thus f0 = 0 since deg(f0) < deg(f). (cid:3) 9 Corollary 0.36. Suppose that F is a field of characteristic zero. If f(x) ∈ F[x] is an irreducible polynomial, then f(x) is separable. Proof. Since F has characteristic zero, and f(x) has positive degree, f0(x) 6= 0. (cid:3) Corollary 0.37. If K is a splitting field over a field F of characteristic zero of a polyno- mial f(x) ∈ F[x], then K is the splitting field of a separable polynomial over F. Proof. Let g be the product of the distinct irreducible factors of f in F[x]. Then K is the splitting field of g(x). The irreducible factors of g(x) are separable by Corollary 0.36. Since the irreducible factors of g must be relatively prime (in F[x] and K[x]), they have no common roots. Thus g is separable (cid:3) Example 0.38. Let F = Z (t) where p is a prime number and t is an indeterminate. p Let f(x) = xp−t ∈ F[x]. f(x) is irreducible but f(x) is not separable over F, as f(x) = √ √ (x− p t)p. A splitting field of f over F is K = F[ p t]. [K : F] = p, but Aut K = {id}. F Theorem0.39. SupposethatN isthesplittingfieldofaseparablepolynomialf(x) ∈ F[x]. Then |Aut N| = [N : F]. F Proof. By the Extension Theorem (Theorem 0.27), the identity automorphism of F ex- tends to [N : F] distinct automorphisms of N, since f(x) has no multiple roots. (cid:3) Definition 0.40. Let G be a finite group of automorphisms of a field K. Define the fixed field KG of G by KG = {a ∈ K | σ(a) = a for all a ∈ K}. The fact that KG is a field follows from the facts that K is a field and G is a group of automorphisms. The fixed field of Aut K necessarily contains F. F Definition 0.41. Suppose that F is a subfield of a field K and K is finite over F (that is [K : F] < ∞). We say that K is Galois over F (or K/F is Galois) if F is the fixed field of Aut K. F If F is a subfield of a field K and L is the fixed field of Aut K, then K is Galois over L. F Theorem 0.42. Suppose that F is a subfield of a field N. Then N is the splitting field of a separable polynomial f(x) ∈ F[x] if and only if N/F is Galois. Proof. First suppose that N is a splitting field of a separable polynomial f(x) ∈ F[x]. Let K be the fixed field of Aut N. Then N is a splitting field of F over K and f(x) is F separable over K also. By applying Theorem 0.39 to both subfields F and K of N, we have |Aut N| = [N : F] = [N : K][K : F] = |Aut N|[K : F]. F K Since Aut N = Aut N, we have [K : F] = 1 and thus K = F, since 1 is then a basis of F K K as a vector space over F. Now suppose that N is Galois over F. Let G = Aut N. N = F[u ,...,u ] for some F 1 k u ∈ N. For 1 ≤ i ≤ k, let a ,...,a be the distinct elements of the set i i,1 i,λi S = {σ(u ) | σ ∈ G}. i i Any two of these sets S are either disjoint or equal. After reindexing, we may assume i that the first n sets S ,...,S are pairwise disjoint, and every S for 1 ≤ i ≤ k is one of 1 n i 10