## Cubic Polynomial $x^3-2$ is Irreducible Over the Field $\Q(i)$

## Problem 399

Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.

Add to solve laterProve that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.

Add to solve laterLet $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements.

For any nonzero element $a\in \F_p$, prove that the polynomial

\[f(x)=x^p-x+a\]
is irreducible and separable over $F_p$.

(Dummit and Foote “Abstract Algebra” Section 13.5 Exercise #5 on p.551)

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