Galois Theory: A Study of Cyclotomic Field ... - Scripps College

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10 Galois Theory L an L-automorphism, which implies G(K/L) ⊃ G. We have containment in both directions, which means G = G(K/L). Since K/F was chosen to be a Galois extension, the order of G is equal to the degree of the extension K over F , or |G| = [K : F ]. If we now apply theorem 3, we know that the degree of the extension of K over L is divisible by the order of G. Since our original hypothesis was that F is contained in L, we are led to the conclusion that [K : F ] = [K : L], which implies F = L. Thus, F must also be a fixed field. □ This corollary leads us into the following theorem relating fixed fields and splitting fields. Theorem 5. Let K be a splitting field of a polynomial f(x) whose coefficients are in the field F . Then K is a Galois extension of F . The reverse direction is also true; every Galois extension is a splitting field of some polynomial f(x) over F . Proof. See [A, p.541] Corollary 6. Every finite field extension is contained in a Galois extension. Proof. Let K/F be a finite extension. Suppose the generators of K over F are α 1 , . . . , α n . We need an irreducible polynomial for each α i over F , call it f i (x). Let the product f 1 · · · f n be f. Then, let M be a field extension of K that is also a splitting field for f over K. This is done by adjoining any roots of some polynomial f i (x) to K if it was not already in K. If M is a splitting field for K, then it must also be a splitting field for F . By theorem 5, M is a Galois extension. □ Corollary 7. Let K/F be a Galois extension. Suppose L is an intermediate field F ⊂ L ⊂ K. Then K/L is also a Galois extension. Proof. Let K be the splitting field of the polynomial f(x) over F . Since
The Main Theorem 11 L ⊂ K, we know K is also a splitting field of f(x) over L. Then by theorem 5, K is a Galois extension of L. □ A critical part in the study of Galois theory is finding the intermediate fields L of an extension K/F . That is, such that F ⊂ L ⊂ K. We will be able to conclude from the Main Theorem of Galois theory that the intermediate fields of a Galois extension are in a bijective correspondence with the subgroups of the Galois group. 3.2 The Main Theorem We will now return our attention to the Main Theorem of Galois Theory. Definition. Given an extension K/F , we will call any field L with F ⊂ L ⊂ K an intermediate field. Theorem 8 (The Main Theorem of Galois Theory). If K is a Galois extension of a field F , and if G = G(K/F ) is its Galois group, then the function given by H ↦→ K H is a bijective map from the set of subgroups of G to the set of intermediate fields L. Its inverse function is defined by L ↦→ G(K/L). This corresponds to the property that if H = G(K/L), then [K : L] = |H|, hence the degree of the extension [L : F ] is equal to the index of the subgroup [G : H].
Page 1: Galois Theory: A Study of Cyclotomi
Page 5: Contents Abstract Acknowledgments i
Page 9: List of Tables 4.1 Field Generator
Page 13: Chapter 1 Introduction Galois theor
Page 16 and 17: 4 The Basics of Field Theory Defini
Page 19 and 20: Chapter 3 Galois Theory 3.1 Introdu
Page 21: Introduction 9 is, L = L ′ = K. T
Page 25 and 26: Proof of the Main Theorem of Galois
Page 27: Intermediate Galois Extensions 15 i
Page 30 and 31: 18 Cyclotomic Field Extensions grou
Page 32 and 33: 20 Cyclotomic Field Extensions a =
Page 34 and 35: 22 Cyclotomic Field Extensions Then
Page 36 and 37: 24 Cyclotomic Field Extensions Q(ζ
Page 38 and 39: 26 Cyclotomic Field Extensions Q co
Page 40 and 41: 28 Cyclotomic Field Extensions that
Page 42 and 43: 30 Cyclotomic Field Extensions sion
Page 45: Bibliography Artin, M. (1991). Alge

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