The degree of extension

Question

I have two extensions and must find their degrees: a) $\mathbb{C}:\mathbb{Q}$; b) $\mathbb{R\{5}\}:\mathbb{R}$. I know that a) degree is infinity and b) is 1. It for me seems trivial, but how it explain in mathematical way? Maybe somebody knew simple definition of degree of extension?

Answer 1

The degree of an extension $E/F$ is by definition the dimension of $E$ as a vector space over $F$. For example, if we look at $\mathbb{C}/\mathbb{Q}$ then the degree is obviously infinite. Why? Well, suppose $\mathbb{C}$ is a finite dimensional vector space over $\mathbb{Q}$. Then there is a finite set $\{z_1,...,z_n\}\subseteq\mathbb{C}$ such that $\mathbb{C}=\{q_1z_1+q_2z_2+...+q_nz_n: q_1,q_2,...,q_n\in\mathbb{Q}\}$. But that means $\mathbb{C}$ is countable which is of course a contradiction.