I want to find a cubic extension of $\mathbb{Q}$

Question

I want to find a cubic extension of $\mathbb{Q}$. I think $\mathbb{Q}(\sqrt{2}, \sqrt{3},\sqrt{5})$ is an example. Is this right?

Answer 1

If $\alpha$ is algberaic over $\mathbb{Q}$ with minimal polynomial of degree $3$ then take $\mathbb{Q}[\alpha]$, for example $$\mathbb{Q}[\sqrt[3]{2}]$$

Answer 2

Note that $Q(\alpha)$ is an extension of degree $k$ if the minimal polynomial of $\alpha$ over $\mathbb{Q}$ has degree $k$. Hence, $Q(\sqrt{2})$ is an extension of degree $2$ of $\mathbb{Q}$. This implies that every number field containing $Q(\sqrt{2})$ will have a degree even over $\mathbb{Q}$. Hence, if you want cubic extension of $\mathbb{Q}$, you can take $\mathbb{Q}[\omega]$ where $\omega$ is a root of $x^3-2$ for instance. Or any polynomial of degree $3$, irreducible over $\mathbb{Q}$ (i.e. with no root, since the degree is $3$).