Example 1

In the field of complex numbers $\mathbf{C}$, the number $i$ is algebraic of degree 2 over the prime field $\mathbf{Q}$; for if $f(X) = X^2 + 1$, then $f(i) = 0$, and $x^2 + 1 \neq 0$ for all $x \in \mathbf{Q}$, hence $i \notin \mathbf{Q}$. The field $\mathbf{Q}(i)$ is thus an extension of degree 2 of $\mathbf{Q}$; it consists of all numbers $a + bi$, where $a, b$ are rational. Likewise $i$ is algebraic of degree 2 over the field $\mathbf{R}$ of real numbers, and $\mathbf{C}$ is an extension of degree 2 of $\mathbf{R}$. \*