Tagged: cubic polynomial

Problem 399

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

Problem 335

Consider 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 $\Q$.