Notation issue with 'Show that $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$.'

Question

Show that $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$.

I haven't seen this notation $\mathbb{Q}[\sqrt{2}]$ before. Can someone clarify what it means?

Answer 1

$\mathbb{Q}[\sqrt{2}]$ is the smallest field which contains $\mathbb{Q}$ and $\sqrt{2}$

The axioms of fields dictate

$\mathbb{Q}[\sqrt{2}]=\{{a+b\sqrt{2}:a,b\in\mathbb{Q}}\}$

Answer 2

Expanding on @Omnomnomnom's comment, $\Bbb Q[\sqrt{2}]$ denotes the set of values $p(\sqrt{2})$ can take for $p$ a $1$-variable polynomial function whose coefficients are in $\Bbb Q$. Since $\sqrt{2}$ is a root of $x^2-2$, this is equivalent to writing $\Bbb Q[\sqrt{2}]=\Bbb Q+\sqrt{2}\Bbb Q$. You can probably spot a basis of this $2$-dimensional vector space now. Similarly, $\Bbb Q[\sqrt[3]{2}]$ is of dimension $3$.