Do all elements in a quadratic extension of $\mathbb Q$ have degree less than or equal to $2$ over $\mathbb Q$?

Question

Do all elements in a quadratic extension of $\mathbb Q$ have degree less than or equal to $2$ over $\mathbb Q$?

For example, do all elements in $\mathbb Q(\sqrt{2})$ satisfy a degree $2$ polynomial over $\mathbb Q$?

Answer 1

Yes. For any field $K$ and any $\alpha$ algebraic over $K$, the degree of $\alpha$ is the dimension of $K[\alpha]$ over $K$.

It follows that if $K\subseteq L$ is an algebraic extension and $\alpha\in L$, then $[L:K]=[L:K(\alpha)][K(\alpha):K]$. In particular, $[K(\alpha):K]$ divides $[L:K]$.