TRACE AND NORM, II

Let L/K be a finite Galois extension, with Galois group G = Gal(L/K). We can express characteristic polynomials, traces, and norms for the extension L/K in terms of G. Theorem 1.1. When L/K is a finite Galois extension with Galois group G and α ∈ L, χα,L/K(X) = ∏ (X − σ(α)). In particular, TrL/K(α) = ∑ σ(α), σ∈G σ∈G NL/K(α) = ∏ σ(α). Proof. Let πα,K(X) be the minimal polynomial of α over K, so χ α,L/K(X) = πα,K(X) n/d, where n = [L: K] and d = [K(α) : K] = deg πα,K. From Galois theory, πα,K(X) = d∏ (X − σi(α)), i=1 where σ1(α),..., σd(α) are all the distinct values of σ(α) as σ runs over the Galois group. For each σ ∈ G, σ(α) = σi(α) for a unique i from 1 to d. Moreover, σ(α) = σi(α) if and only if σ ∈ σiH, where H = {τ ∈ G: τ(α) = α} = Gal(L/K(α)). Therefore as σ runs over G, the number σi(α) appears as σ(α) whenever σ is in the left coset σiH, so σi(α) occurs |H | times, and |H | = [L: K(α)] = [L: K]/[K(α) : K] = n/d. Therefore d∏ (X − σ(α)) = (X − σi(α)) n/d ()n/d d∏ = (X − σi(α)) = πα,K(X) n/d, σ∈G i=1 and that power of the minimal polynomial is the characteristic polynomial. Example 1.2. In Q ( √ d)/Q, where d is a nonsquare rational number, the two elements of the Galois group are σ(a + b √ d) = a + b √ d and σ(a + b √ d) = a − b √ d. Then i=1 σ∈