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3. Let f ∈ Z[x] be the minimal polynomial of a over Z. Suppose that p [OK : Z[a]] is a prime. 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.

Xt |) for 1 ≤ j ≤ t. , in polynomial time in the number of digits of the input. The current optimal implementation (and practically optimal algorithms) for computing LLL reduced basis are due to Damien Stehle, and are included standard in Magma in Sage. Stehle’s code is amazing – it can LLL reduce a random lattice in Rn for n < 1000 in a matter of minutes!! 0494159999999999 There is even a very fast variant of Stehle’s implementation that computes a basis for L that is very likely LLL reduced but may in rare cases fail to be LLL reduced.

10), OK is noetherian. If R is an integral domain, the field of fractions Frac(R) of R is the field of all equivalence classes of formal quotients a/b, where a, b ∈ R with b = 0, and a/b ∼ c/d if ad = bc. For example, the field √ of fractions of √ Z is (canonically isomorphic to) Q and the field of fractions of Z[(1 + 5)/2] is Q( 5). 19). 1. We compute the fraction fields mentioned above. sage: Frac(ZZ) Rational Field In Sage the Frac command usually returns a field canonically isomorphic to the fraction field (not a formal construction).

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