5 – Simple extensions

(This is part of my series Field theory for high-schoolers.)

This post describes a particular type of field extensions called simple extensions, thus named because they can be constructed from the base field in a simple and systematic manner.

Let $f(x)$ be a non-constant polynomial with real coefficients that has no root in $\mathbf{R}$. Kronecker’s theorem tells us that there exists some extension of $\mathbf{R}$ in which $f(x)$ has a root, but it tells us nothing about what that extension actually is. In other words, it doesn’t tell us how to construct such an extension. By “constructing” a field extension, we mean being able to describe its elements.

The aim of this series is to illustrate how $\mathbf{C}$ can be constructed from $\mathbf{R}$. This implies that we should know how to construct an extension of some known field. This post and the next will discuss how to construct a field extension by adjoining to a base field a single element of a bigger field, and show how $\mathbf{C}$ can be constructed from $\mathbf{R}$ in that way.

5.1 Simple extensions

Let $F$ be a field, and $E$ be an extension of $F$. For some element $\alpha$ of $E$, we can construct the field $F(\alpha)$, which is defined as the smallest field containing both $F$ and $\alpha$. Such an extension $F(\alpha)$, created from $F$ by adjoining to it exactly one element of some known extension of $F$, is called a simple extension of $F$.

In general, just adding the element $\alpha$ to $F$ will break the field structure. For example if we just add the element $i$ to $\mathbf{R}$, the field structure is broken because for example the operation $1 + i$ does not yield an element of our new set. Thus our field $\mathbf{R}(i)$ must also contain the element $1+i$, and so on. Since we know at least one field containing $\mathbf{R}$ and $i$ (the field $\mathbf{C}$), it seems reasonable to expect that after having added all the necessary elements, we will obtain a field containing both $\mathbf{R}$ and $i$.

Given a field $F$ and an element $\alpha$ of some extension $E$ of $F$, it is always possible to construct $F(\alpha)$, the smallest field containing both $F$ and $\alpha$. In some cases it will be $E$ itself, but not always. How exactly to construct $F(\alpha)$ depends on whether $\alpha$ is algebraic or transcendental over $F$
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If $\alpha$ is algebraic over $F$, the extension $F(\alpha)$ is said to be an algebraic extension of $F$. If $\alpha$ is transcendental over $F$, it is a transcendental extension of $F$. As you might expect, in algebra we are mostly concerned with algebraic extensions, and thus we will not discuss transcendental extensions here. The next post will describe how to construct simple algebraic extensions.

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