Determine the splitting field and its degree over $\mathbb{Q}$ for $x^4 + x^2 + 1$.

#### Solution

Replacing $x^2$ with $y$ and applying the quadratic formula yields

Substituting back $y$ with $x^2$ and equating both terms to zero gives

Therefore $x^4 + x^2 + 1$ splits completely in $\mathbb{Q}(\sqrt{-3})$ which, by the reasoning in 13.4.1, is an extension over the rationals of degree 2.