## Inessential Discriminant Divisors

Definition 4.3.1   A prime is an inessential discriminant divisor if for every .

See Example 6.2.6 below for why it is called an inessential discriminant divisor'' instead of an inessential index divisor''.

Since is the absolute value of , where is the characteristic polynomial of , an inessential discriminant divisor divides the discriminant of the characteristic polynomial of any element of .

Example 4.3.2 (Dedekind)   Let be the cubic field defined by a root of the polynomial . We will use SAGE to show that  is an inessential discriminant divisor for .
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8); K
Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8
sage: K.factor_integer(2)
(Fractional ideal (1/2*a^2 - 1/2*a + 1)) *
(Fractional ideal (a^2 - 2*a + 3)) *
(Fractional ideal (3/2*a^2 - 5/2*a + 4))

Thus , with the distinct, and one sees directly from the above expressions that for each . If for some with minimal polynomial , then must be a product of three distinct linear factors, which is impossible, since the only linear polynomials in are and .

William Stein 2012-09-24