Find splitting field of x^4 -x^2 -2 over Z3?

Answer 1

Hmmm. Well over Z3, x^4 - x^2 - 2 = x^4 + 2x^2 + 1 = (x^2 + 1)^2, so the splitting field of x^4 - x^2 - 2 over Z3 will be the same as the splitting field of x^2 + 1 over Z3, namely Z3 adjoined a 2nd root of -1. Explicitly, take the polynomial ring Z3[X] and mod out by the ideal generated by X^2 + 1. Since this polynomial is irreducible over Z3, it generates a maximal ideal and hence the quotient Z3[X]/(X^2 + 1) is a field where x^2 + 1 = 0.

So I guess that is the splitting field: Z3[X] / (X^2 + 1).