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 .

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

William Stein 2012-09-24