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
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
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