📂Abstract Algebra

Simple Extension Field

Definition¹

If an extension field $E$ of $F$ satisfies $E = F( \alpha )$ for some $\alpha \in E$, then $E$ is called a simple extension of $F$.

Explanation

Simply put, $F ( \alpha )$ can be seen as an extension obtained by adding just one—hence simple—element $\alpha$ that was not in $F$. Speaking of the field of real numbers $\mathbb{R}$, if we adjoin $i \in \mathbb{C}$ from its extension field $\mathbb{C}$, we get $\mathbb{R} ( i ) = \mathbb{C}$.

An important fact is that if $E = F ( \alpha )$ for $\alpha \in E$, then every $\beta \in E$ is uniquely represented as $$ \beta = b_{0} + b_{1} \alpha + \cdots + b_{n} \alpha^n $$ Here $\left\{ b_{k} \right\}_{k =1}^{n}$ are elements of $F$; considering the field of complex numbers as a simple extension of the field of real numbers, one can easily check that every complex number $z \in \mathbb{C}$ is represented, for some $x , y \in \mathbb{R}$, as $$ z = x_{0} + y_{0} i + x_{1} i^2 + y_{1} i^3 + \cdots = x + i y $$

Meanwhile, as interesting examples of simple extensions, one can consider the Gaussian integers $\mathbb{Z} [i]$ and the Eisenstein integers $\mathbb{Z} [\omega]$, obtained by adjoining the complex numbers $i$ and $\omega$ to the integers.