By Robin Chapman

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**Additional info for Algebraic Number Theory: summary of notes [Lecture notes]**

**Example text**

Here P1 = P2 and both P1 and P2 have norm p. In this case we say that p splits in K. 3. X 2 − m splits into two equal factors over Fp . This happens only when p = 2 or when p | m. When p = 2 then X 2 − m ≡ X 2 or (X + 1)2 (mod 2) according to the parity of m. When p | m then X 2 − m ≡ X 2 √ (mod p). Then p√= P 2 where P = p, m , unless p = 2 and m is odd when P = 2, m + 1 . In any case P has norm p. In this case we say that p ramifies in K. Note that p ramifies if and only if p divides the discriminant of K.

3 Let G be a free abelian group of rank n, and let H be a subgroup of G. Then H is free abelian of rank m where m ≤ n. Proof We may assume that G = Zn . We begin by defining various subgroups H1 , . . , Hn of H. We let H1 = H and for j > 1 let Hj = {(0, 0, . . , 0, cj , cj+1 , . . , cn ) : cj , cj+1 , . . , cn ∈ Z} ∩ H. That is Hj is the set of all vectors in H where the first j − 1 entries vanish. We also let Kj = {cj : (0, 0, . . , 0, cj , cj+1 , . . , cn ) ∈ Hj } for each j. That is, Kj is the set of j-th entries of vectors in Hj .

We denote the ideal class containing the fractional ideal I by [I]. We can consider ClK as the set of such symbols [I] obeying the rule that [I] = [J] whenever J = βI, β ∈ K and β = 0 and with the operation [I][J] = [IJ]. The identity element of ClK is [ 1 ] = [OK ]. In addition since each fractional ideal I of K has the form βJ where J is an ideal, then [I] = [J] and so each ideal class contains ideals. √ √ Example √ Let K = Q( −6) √ so that OK = Z[ −6]. Consider the ideals I = 2, −6 and J = 3, −6 .