## 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 laterConsider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$.

Let $\alpha$ be any real root of $f(x)$.

Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.