📂Abstract Algebra

Simple Enlargement Body

Definition ¹

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 expansion by adding just one $\alpha$ that was not in $F$. Speaking in terms of the field of real numbers $\mathbb{R}$, adding $i \in \mathbb{C}$ to its extension field $\mathbb{C}$ results in $\mathbb{R} ( i ) = \mathbb{C}$.

An important fact is that for $\alpha \in E$, if $E = F ( \alpha )$, then all $\beta \in E$ are $$ \beta = b_{0} + b_{1} \alpha + \cdots + b_{n} \alpha^n $$ uniquely represented like this. In this case, $\left\{ b_{k} \right\}_{k =1}^{n}$ is an element of $F$, and thinking about the field of complex numbers as a simple extension of real numbers, it’s easy to see that all complex numbers $z \in \mathbb{C}$ can be represented for some $x , y \in \mathbb{R}$ as $$ z = x_{0} + y_{0} i + x_{1} i^2 + y_{1} i^3 + \cdots = x + i y $$

On another note, an interesting example of simple extensions includes Gaussian integers $i$ and $\omega$ added to integers, such as Gaussian integers $\mathbb{Z} [i]$ and Eisenstein integers $\mathbb{Z} [\omega]$.