Why the concept of a field extension is a natural one

• Discovering the notion of field extensions.

• It is not the aim of this page to get at all far with Galois theory. All I shall demonstrate is that one can be led very naturally to think about field extensions, even if one does not have the mentality of an algebraist.

• A well known elementary exercise is to show that $2^{1/2}+3^{1/2}$ is algebraic. The answer is as follows: $x^2={\left(2^{1/2}+3^{1/2}\right)}^2=2+3+2\cdot 6^{1/2}$. It follows that ${\left(x^2-5\right)}^2=24$, which shows that x is algebraic.

  • $$x^2-5=2\ast 6^{\frac{1}{2}}$$

• It is now very natural to look at other examples, or even to try to prove a general theorem. What can we say about $2^{1/3}+3^{1/3}$, for example? If we square it, we obtain $2^{2/3}+2.6^{1/3}+3^{2/3}$, which doesn’t look very nice. On the other hand, cubing would turn at least some of the terms into integers, so let’s try that. We obtain $2+3.2^{2/3}{3}^{1/3}+3.2^{1/3}{3}^{2/3}+3$. Unfortunately, this is still not very nice, because there are two irrational terms. In other words, we don’t seem to be any better off than we were when we started.

  Let us now apply the following general principle.