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

> K<a> := NumberField(x^3 + x^2 - 2*x + 8); > OK := MaximalOrder(K); > Factorization(2*OK); [ <Prime Ideal of OK Basis: [2 0 0] [0 1 0] [0 0 1], 1>, <Prime Ideal of OK Basis: [1 0 1] [0 1 0] [0 0 2], 1>, <Prime Ideal of OK Basis: [1 0 1] [0 1 1] [0 0 2], 1> ]Thus , with the distinct. Moreover, one can check that . If for some with minimal polynomial , then must be a product of three

William Stein 2004-05-06