Meaning of $[\mathbb K : \mathbb Q]$

Question

What is the meaning of $[\mathbb K : \mathbb Q]$ where $\mathbb K$ and $\mathbb Q$ are fields. This is galois theory, abstract algebra.

What does this actually mean?

Answer 1

This means the dimension of $\mathbb{K}$ as a $\mathbb{Q}$-vector space. It is called the degree of the extension.

If $\mathbb{K} = \mathbb{Q}(\alpha)$, for an algebraic $\alpha$, then the degree of the extension is the degree of the minimal polynomial. Indeed, then $\alpha^{i}$ for $i = 0, \dots, [\mathbb K : \mathbb Q]-1$ is a basis for $\mathbb K$ as a vector space over $\mathbb{Q}$