## Problem 399

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

## Problem 230

Let $\Q$ be the field of rational numbers.

(a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$?

(b) Find the Galois group of $f(x)$ over $\Q$.

## Problem 99

Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.

## Problem 92

Determine the splitting field and its degree over $\Q$ of the polynomial
$x^4+x^2+1.$ Read solution

## Problem 82

Show that the polynomial $x^3-\sqrt{2}$ is irreducible over the field $\Q(\sqrt{2})$.