Let K be a field containing F as a subfield, written as $K/F$. Call $F$ the base field of $K/F$ and $K$ the extension field of $K/F$.
Definition: Let $K/F$ be an extension of fields. K is a vector space over $F$ by restriction of scalars. Denote the dimension, $\text{dim}_F K$, of $K$ as an F-vector space by $[K:F]$ and call it the degree of $K/F$. We call $K/F$ a finite extension if $[K:F]$ is finite, and an infinite extension otherwise.
This is copied from an Abstract Algebra Lecture Notes. It gives a few examples to show this definition.
Example: 1). $F/F$ iss a finite extension of degree one; 2).$\mathbb{C}/\mathbb{R}$ is a finite extension of degree two; 3). $\mathbb{R}/\mathbb{Q}$ is an infinite extension.
I am not sure how to verify those examples. Specifically why is $\text{dim}_F F$ of degree 1, $\text{dim}_{\mathbb{R}}\mathbb{C}$ of degree 2 and such? Need help with walking through the idea of checking those examples. Appreciate it!