Construct field $\mathbb{Q}(\alpha)$

Question

There is defined field K and polynomial $m \in K[x]$ which is irreducible over K. $K=\mathbb{Q} , m(x) = x^{3} - 2$. I have to choose on of the root $\alpha$ of polynomial $m$ and construct field $\mathbb{Q}(\alpha)$. Root should be from $\mathbb{C}$. Let's say I chose root $- \sqrt[3]{-2}$. How could I construct that field?

Answer 1

Since your polynomial is irreducible, the field is just $$\mathbb Q(\alpha )=\{a+b\alpha +c\alpha ^2\mid a,b,c\in\mathbb Q\}.$$ Indeed, $$\mathbb Q(\alpha )\cong \mathbb Q[X]/(X^3-2)\cong \{a+b\alpha +c\alpha ^2\mid a,b,c\in\mathbb Q\}.$$