By Matthew Baker
Path Notes (Fall 2006) Math 8803, Georgia Tech, model 24 Nov 2012
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Extra resources for Algebraic Number Theory [Course Notes]
Let ∆ := ∆(α1 , . . , αn ), ∆ := ∆(α1 , . . , αn ). Writing each basis in terms of the other, we find that there are nonsingular n×n matrices M, M with integer coefficients such that ∆ = det(M )2 ∆ and ∆ = det(M )2 ∆. The result follows easily from this. 2, we may define the discriminant of OK (or, by abuse of terminology, the discriminant of K) to be the discriminant of any integral basis for OK . We write ∆(OK ) or ∆K for the discriminant of OK . 3. If α1 , . . , αn ∈ OK form a basis for K/Q and M denotes the Z-module spanned by α1 , .
We will sometimes write Km to denote the cyclotomic field Q(ζm ). We recall for later use that Km /Q is a Galois extension with Galois group isomorphic to (Z/mZ)∗ . The fact that Km /Q is Galois follows t from the fact that, by the irreducibility of Φm (x), ζm ∈ Km is a root of Φm (x) whenever 1 ≤ t < m and gcd(t, m) = 1. Furthermore, the map from (Z/mZ)∗ to Gal(Km /Q) sending t to the automorphism t σt : ζm → ζm is a group isomorphism. 30. Let m be a positive integer, and let d = φ(m) be the degree of the minimal polynomial Φm (x) of ζm over Q.
Proof. 15, we have OK /pi = OK /(p, gi (θ)) ∼ = Z[θ]/ (p, gi (θ)) ∼ = Z[x]/ (p, gi (x)) ∼ = Fp [x]/ gi (x) . Since gi (x) is irreducible over Fp , we know that Fp [x]/ gi (x) is a field of degree fi := deg(gi ) over Fp . Therefore pi is a prime ideal with norm pfi . 1) ei fi = deg g(x) = n. i=1 3. THE SPLITTING OF PRIMES 41 We now prove that the pi ’s are distinct. Given i = j, we know that gi (x) and gj (x) are relatively prime in Fp [x]. So there exist a(x), b(x) ∈ Z[x] such that 1 = a(x) · gi (x) + b(x) · gj (x) in Fp [x].