The cover is easy to understand because it is defined by the single equation , in the sense that . To give a maximal ideal of such that is the same as giving a homomorphism up to automorphisms of the image, which is in turn the same as giving a root of in up to automorphism, which is the same as giving an irreducible factor of the reduction of modulo .

Let denote a fixed algebraic closure of ; thus is an algebraically closed field of characteristic , over which all polynomials in factor into linear factors. Any homomorphism sends to 0, so is the composition of a homomorphism with a homomorphism . Since , the homomorphisms are in bijection with the homomorphisms . The homomorphisms are in bijection with the roots of the reduction modulo of the minimal polynomial of in .

As suggested in the proof of the lemma, we find all homomorphisms by finding all homomorphism . In terms of ideals, if is a maximal ideal of , then the ideal of is also maximal, since

We formalize the above discussion in the following theorem (note: we will not prove that the powers are here):

We return to the example from above, in which
, where is
a root of
. According to *SAGE*, the ring
of integers has discriminant
:

sage: K.<a> = NumberField(x^5 + 7*x^4 + 3*x^2 - x + 1) sage: D = K.discriminant(); D 2945785 sage: factor(D) 5 * 353 * 1669The order has the same discriminant as , which is the same as the discriminant of , so and we can apply the above theorem. (Here we use that the index of in is the square of the quotient of their discriminants, a fact we will prove later in Section 6.2.)

sage: R.<x> = QQ[] sage: discriminant(x^5 + 7*x^4 + 3*x^2 - x + 1) 2945785We have

If we replace by , then the index of in will be a power of , which is coprime to , so the above method will still work.

sage: K.<a> = NumberField(x^5 + 7*x^4 + 3*x^2 - x + 1) sage: f = (7*a).minpoly('x') sage: f x^5 + 49*x^4 + 1029*x^2 - 2401*x + 16807 sage: f.disc() 235050861175510968365785 sage: factor(f.disc() / K.disc()) 7^20 sage: f.factor_mod(5) (x + 4) * (x + 1)^2 * (x^2 + 3*x + 3)Thus factors in as

sage: K.<a> = NumberField(x^5 + 7*x^4 + 3*x^2 - x + 1) sage: f = (5*a).minpoly('x') sage: f x^5 + 35*x^4 + 375*x^2 - 625*x + 3125 sage: f.factor_mod(5) x^5

William Stein 2012-09-24