Field Extensions and the Basic Theory of Galois Fields
The present chapter is devoted to the basic theory of finite fields, including existence and uniqueness theorems as well as the main structural results. For this purpose, we also extend the fundamental material covered in Chapters 1 and 2 by proving sev
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Field Extensions and the Basic Theory of Galois Fields
Abstract The present chapter is devoted to the basic theory of finite fields, including existence and uniqueness theorems as well as the main structural results. For this purpose, we also extend the fundamental material covered in Chapters 1 and 2 by proving several results on field extensions in general (in particular, in the first two sections).
3.1 The Splitting Field of a Polynomial Let F be an arbitrary field and f (x) any non-constant polynomial over F. The main result of the present section guarantees the existence of a smallest extension field E over F such that f (x) splits over E into linear factors. Moreover, this field E has finite degree and is unique up to isomorphism, which motivates calling E the splitting field of f over F. We start with the following basic result on extensions with finite degree: Lemma 3.1.1. Let E/F be a field extension, and let K be an intermediate field of E/F. Then E/F has finite degree if and only if both E/K and K/F have finite degree. In this case, [E : F] = [E : K] · [K : F]. Proof. If E/F has finite degree, then E/K has finite degree, since F ⊆ K and any F-basis of E generates E as a K-vector space; and K/F has finite degree since K is an F-subspace of E. Now assume that both E/K and K/F have finite degree, and let B be an F-basis for K and C a K-basis for E. We claim that the product set BC := {bc : b ∈ B, c ∈ C} is a basis for E over F. This will establish the assertion in view of |BC| = |B × C|, which holds as the mapping B ×C → BC, (b, c) → bc is easily seen to be injective. (Assume bc = b c , that is, bc−b c = 0. If c = c , then b = b = 0, since b, b ∈ K and © Springer Nature Switzerland AG 2020 D. Hachenberger and D. Jungnickel, Topics in Galois Fields, Algorithms and Computation in Mathematics 29, https://doi.org/10.1007/978-3-030-60806-4_3
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3 Field Extensions and the Basic Theory of Galois Fields
C is a K-basis of E, contradicting the linear independence of K over F. Therefore c = c , and b = b .) We now check that BC is linearly independent over F. Assume ∑bc∈BC λbc bc = 0, where all λbc ∈ F, that is, ∑ ∑ λbc b c = 0. c∈C
b∈B
As the inner sums are elements from K, the linear independence of C over K shows ∑b∈B λbc b = 0 for all c ∈ C. Then the linear independence of B over F gives λbc = 0 for all b ∈ B and all c ∈ C, as claimed. Finally, consider an arbitrary element v ∈ E. Then there are scalars αc in K with v = ∑c∈C αc c. For each αc , there exist scalars βb,c ∈ F such that αc = ∑b∈B βb,c b. This gives v = ∑b∈B, c∈C βb,c bc, and therefore BC generates E as F-vector space. Thus BC is indeed a basis for E over F, and E/F has degree [E : F] = |BC| = [E : K] · [K : F]. It is worthwhile to state the following fact established in the proof of Lemma 3.1.1 explicitly: Corollary 3.1.2. Let K be an intermediate field of a field extension E/F with finite degree, and assume that B is an F-basis for K and C a K-basis for E. Then BC is an F-basis for E. We next show that any non-constant polynomial f (x
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