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An Introduction to Galois Module Structure

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An Introduction to Galois Module Structure AN INTRODUCTION TO GALOIS MODULE STRUCTURE Nigel Byott University of Exeter Omaha, May 2019 Its Galois group is Gal(N=K) := ffield automorphisms σ : N ! N with σ(k) = k 8k 2 Kg: Then Gal(N=K) is a group of order n. The Fundamental Theorem of Galois Theory says there is a bijection between subgroups of Gal(N=K) and fields E with K ⊆ E ⊆ N. The Nomal Basis Theorem says there is an element α 2 N (called a normal basis generaor) such that fσ(α): σ 2 Gal(N=K)g is a basis for N as a K-vector space. Thus every β 2 N can be written in a unique way as X β = cσσ(α); cσ 2 K: σ Galois Theory: A Quick Summary An extension N=K of fields of (finite) degree n = [N : K] is Galois if it is normal and separable. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 2 / 25 The Fundamental Theorem of Galois Theory says there is a bijection between subgroups of Gal(N=K) and fields E with K ⊆ E ⊆ N. The Nomal Basis Theorem says there is an element α 2 N (called a normal basis generaor) such that fσ(α): σ 2 Gal(N=K)g is a basis for N as a K-vector space. Thus every β 2 N can be written in a unique way as X β = cσσ(α); cσ 2 K: σ Galois Theory: A Quick Summary An extension N=K of fields of (finite) degree n = [N : K] is Galois if it is normal and separable. Its Galois group is Gal(N=K) := ffield automorphisms σ : N ! N with σ(k) = k 8k 2 Kg: Then Gal(N=K) is a group of order n. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 2 / 25 The Nomal Basis Theorem says there is an element α 2 N (called a normal basis generaor) such that fσ(α): σ 2 Gal(N=K)g is a basis for N as a K-vector space. Thus every β 2 N can be written in a unique way as X β = cσσ(α); cσ 2 K: σ Galois Theory: A Quick Summary An extension N=K of fields of (finite) degree n = [N : K] is Galois if it is normal and separable. Its Galois group is Gal(N=K) := ffield automorphisms σ : N ! N with σ(k) = k 8k 2 Kg: Then Gal(N=K) is a group of order n. The Fundamental Theorem of Galois Theory says there is a bijection between subgroups of Gal(N=K) and fields E with K ⊆ E ⊆ N. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 2 / 25 Thus every β 2 N can be written in a unique way as X β = cσσ(α); cσ 2 K: σ Galois Theory: A Quick Summary An extension N=K of fields of (finite) degree n = [N : K] is Galois if it is normal and separable. Its Galois group is Gal(N=K) := ffield automorphisms σ : N ! N with σ(k) = k 8k 2 Kg: Then Gal(N=K) is a group of order n. The Fundamental Theorem of Galois Theory says there is a bijection between subgroups of Gal(N=K) and fields E with K ⊆ E ⊆ N. The Nomal Basis Theorem says there is an element α 2 N (called a normal basis generaor) such that fσ(α): σ 2 Gal(N=K)g is a basis for N as a K-vector space. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 2 / 25 Galois Theory: A Quick Summary An extension N=K of fields of (finite) degree n = [N : K] is Galois if it is normal and separable. Its Galois group is Gal(N=K) := ffield automorphisms σ : N ! N with σ(k) = k 8k 2 Kg: Then Gal(N=K) is a group of order n. The Fundamental Theorem of Galois Theory says there is a bijection between subgroups of Gal(N=K) and fields E with K ⊆ E ⊆ N. The Nomal Basis Theorem says there is an element α 2 N (called a normal basis generaor) such that fσ(α): σ 2 Gal(N=K)g is a basis for N as a K-vector space. Thus every β 2 N can be written in a unique way as X β = cσσ(α); cσ 2 K: σ Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 2 / 25 Then K[G] is a ring with multiplication ! ! ! X X X X X cσσ dτ τ = cσdτ στ = cσdσ−1ρ ρ. σ τ σ,τ ρ σ We call K[G] the group algebra of G over K. Then K[G] acts on N; for β 2 N we have ! X X cσσ · β = cσσ(β) 2 N: σ σ Thus N becomes a module over the ring K[G]. Reinterpreting the Normal Basis Theorem Write G = Gal(N=K) and let ( ) X K[G] = cσσ : cσ 2 K ; σ2G a K-vector space over K of dimension n. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 3 / 25 We call K[G] the group algebra of G over K. Then K[G] acts on N; for β 2 N we have ! X X cσσ · β = cσσ(β) 2 N: σ σ Thus N becomes a module over the ring K[G]. Reinterpreting the Normal Basis Theorem Write G = Gal(N=K) and let ( ) X K[G] = cσσ : cσ 2 K ; σ2G a K-vector space over K of dimension n. Then K[G] is a ring with multiplication ! ! ! X X X X X cσσ dτ τ = cσdτ στ = cσdσ−1ρ ρ. σ τ σ,τ ρ σ Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 3 / 25 Then K[G] acts on N; for β 2 N we have ! X X cσσ · β = cσσ(β) 2 N: σ σ Thus N becomes a module over the ring K[G]. Reinterpreting the Normal Basis Theorem Write G = Gal(N=K) and let ( ) X K[G] = cσσ : cσ 2 K ; σ2G a K-vector space over K of dimension n. Then K[G] is a ring with multiplication ! ! ! X X X X X cσσ dτ τ = cσdτ στ = cσdσ−1ρ ρ. σ τ σ,τ ρ σ We call K[G] the group algebra of G over K. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 3 / 25 Thus N becomes a module over the ring K[G]. Reinterpreting the Normal Basis Theorem Write G = Gal(N=K) and let ( ) X K[G] = cσσ : cσ 2 K ; σ2G a K-vector space over K of dimension n. Then K[G] is a ring with multiplication ! ! ! X X X X X cσσ dτ τ = cσdτ στ = cσdσ−1ρ ρ. σ τ σ,τ ρ σ We call K[G] the group algebra of G over K. Then K[G] acts on N; for β 2 N we have ! X X cσσ · β = cσσ(β) 2 N: σ σ Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 3 / 25 Reinterpreting the Normal Basis Theorem Write G = Gal(N=K) and let ( ) X K[G] = cσσ : cσ 2 K ; σ2G a K-vector space over K of dimension n. Then K[G] is a ring with multiplication ! ! ! X X X X X cσσ dτ τ = cσdτ στ = cσdσ−1ρ ρ. σ τ σ,τ ρ σ We call K[G] the group algebra of G over K. Then K[G] acts on N; for β 2 N we have ! X X cσσ · β = cσσ(β) 2 N: σ σ Thus N becomes a module over the ring K[G]. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 3 / 25 This means that N is a free K[G]-module of rank 1. [Note that, unlike vector spaces over a field, modules over a ring do not always have a basis, i.e. are not always free. Thus the Normal Basis Theorem gives non-trivial information about the structure of N as a K[G]-module.] The main question of Galois module structure is: Can we find an analogue of the Normal Basis Theorem at the level of integers? Now if α 2 N is a normal basis generator for N=K, then, for each β 2 N, there is a unique λ 2 K[G] with β = λ · α. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 4 / 25 [Note that, unlike vector spaces over a field, modules over a ring do not always have a basis, i.e. are not always free. Thus the Normal Basis Theorem gives non-trivial information about the structure of N as a K[G]-module.] The main question of Galois module structure is: Can we find an analogue of the Normal Basis Theorem at the level of integers? Now if α 2 N is a normal basis generator for N=K, then, for each β 2 N, there is a unique λ 2 K[G] with β = λ · α. This means that N is a free K[G]-module of rank 1. Nigel Byott (University of Exeter) Galois Module Structure Omaha, May 2019 4 / 25 The main question of Galois module structure is: Can we find an analogue of the Normal Basis Theorem at the level of integers? Now if α 2 N is a normal basis generator for N=K, then, for each β 2 N, there is a unique λ 2 K[G] with β = λ · α.
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