By Stein W.
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Additional resources for A Brief Introduction to Classical and Adelic Algebraic Number Theory
Let t f= i=1 e f i i ∈ Fp [x] where the f i are distinct monic irreducible polynomials. Let pi = (p, fi (a)) where fi ∈ Z[x] is a lift of f i in Fp [X]. Then t pei i . pOK = i=1 We return to the example from above, in which K = Q(a), where a is a root of x5 +7x4 +3x2 −x+1. According to Magma, the maximal order OK has discriminant 2945785: > Discriminant(MaximalOrder(K)); 2945785 The order Z[a] has the same discriminant as OK , so Z[a] = OK and we can apply the above theorem. > Discriminant(x^5 + 7*x^4 + 3*x^2 - x + 1); 2945785 We have x5 + 7x4 + 3x2 − x + 1 ≡ (x + 2) · (x + 3)2 · (x2 + 4x + 2) (mod 5), which yields the factorization of 5OK given before the theorem.
We assume that an order O has been given by a basis w1 , . . , wn and that O that contains Z[a]. Each of the following steps can be carried out efficiently using little more than linear algebra over Fp . 5]. 1. 3, we easily factor pO. 2. [Compute radical] Let I be the radical of pO, which is the ideal of elements x ∈ O such that xm ∈ pO for some positive integer m. Using linear algebra over the finite field Fp , we can quickly compute a basis for I/pO. ) 3. [Compute quotient by radical] Compute an Fp basis for A = O/I = (O/pO)/(I/pO).
Thus the rank of L is m = dim(RL), as claimed. Since σ(OK ) is a lattice in V , the volume of V /σ(OK ) is finite. Suppose w1 , . . , wn is a basis for OK . Then if A is the matrix whose ith row is σ(wi ), then | Det(A)| is the volume of V /σ(OK ). ) 63 64CHAPTER 10. 2. Let OK = Z[i] be the ring of integers of K = Q(i). Then w1 = 1, w2 = i is a basis for OK . The map σ : K → C2 is given by σ(a + bi) = (a + bi, a − bi) ∈ C2 . The image σ(OK ) is spanned by (1, 1) and (i, −i). The volume determinant is 1 1 i −i = | − 2i| = 2.