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By Stein W.A.
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Extra resources for Algebraic number theory, a computational approach
By evaluating a special function and get a decimal approxi√ mation α ∈ C to an algebraic number β ∈ Q. 64940734628582579415255223513033238849340192353916 Now suppose you very much want to find the (rescaled) minimal polynomial f (x) ∈ Z[x] of β just given this numerical approximation α. This is of great value even without proof, since often in practice once you know a potential minimal polynomial you can verify that it is in fact right. Exactly this situation arises in the explicit construction of class fields (a more advanced topic in number theory) and in the construction of Heegner points on elliptic curves.
This computes I, since pO ⊂ I. 3. [Compute quotient by radical] Compute an Fp basis for A = O/I = (O/pO)/(I/pO). The second equality comes from the fact that pO ⊂ I. Note that O/pO is obtained by simply reducing the basis w1 , . . , wn modulo p. Thus this step entirely involves linear algebra modulo p. 4. [Decompose quotient] The ring A is isomorphic to the quotient of O by a radical ideal, so it decomposes as a product A ∼ = A1 × · · · × Ak of finite fields. 7. 5. [Compute the maximal ideals over p] Each maximal ideal pi lying over p is the kernel of one of the compositions O → A ≈ A1 × · · · × Ak → Ai .
2. Then A∼ = F2 × F2 × F2 , which as a ring is not generated by a single element, since there are only 2 distinct linear polynomials over F2 [x]. 9 (Factoring a General Prime Ideal). Let K = Q(a) be a number field given by an algebraic integer a as a root of its minimal monic polynomial f of degree n. We assume that an order O has been given by a basis w1 , . . , 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.