Irrational, Algebraic, and Transcendental Numbers

Abstract

We have seen in the last chapter that the field of complex numbers admits no further algebraic extensions. Within the field of complex numbers, however, there are many numbers that are not algebraic over $\mathbb {Q}$. In fact, the algebraically closed field of all algebraic numbers in $\mathbb {C}$ is a countable set, for you can check that the algebraic numbers over $\mathbb {Q}$ that have a minimal polynomial of degree n are countable. Letting n vary gives a countable collection of countable sets, which is therefore countable.