By Stein W.A.

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**Example text**

Let t e f i i ∈ Fp [x] f= i=1 54 CHAPTER 4. FACTORING PRIMES 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 f = x5 + 7x4 + 3x2 − x + 1. discriminant(); D 2945785 sage: factor(D) 5 * 353 * 1669 The order Z[a] has the same discriminant as f (x), which is the same as the discriminant of OK , so Z[a] = OK and we can apply the above theorem.

Wn and that O that contains Z[a]. For any prime p ∈ Z, the following algorithm computes the set of maximal ideals of O that contain p. 1. 3 we factor pO. 3. A GENERAL METHOD 59 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. , I | pO; also I is the product of the primes that divide p, without multiplicity. Using linear algebra over the finite field Fp , we compute a basis for I/pO by computing the abelian subgroup of O/pO of all nilpotent elements.

If not, then there is a nonzero integral ideal I that is maximal among all nonzero integral ideals that do not have an inverse. Every ideal is contained in a maximal ideal, so there is a nonzero prime ideal p such that I ⊂ p. Multiplying both sides of this inclusion by p−1 and using that OK ⊂ p−1 , we see that I ⊂ p−1 I ⊂ p−1 p = OK . If I = p−1 I, then arguing as in the proof that p−1 is an inverse of p, we see that each element of p−1 preserves the finitely generated Z-module I and is hence integral.

### Algebraic number theory, a computational approach by Stein W.A.

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