Definition 1

The field $\mathbf{R}[X]/(X^2 + 1)$ is called the field of complex numbers and is denoted by $\mathbf{C}$. The canonical image of $X$ in $\mathbf{C}$ is denoted by $i$, so that $\mathbf{C}$ is obtained from the field $\mathbf{R}$ by algebraic adjunction of the root $i$ of the polynomial $X^2 + 1$. The elements of $\mathbf{C}$ are called complex numbers.

From an algebraic point of view the importance of the field $\mathbf{C}$ is due to the following fundamental theorem: