## Now there’s completeness and then there’s completeness…

15 October 2009

This post achieves a fortuitous segue from the last post into my serious of articles on the beauty of Galois theory.

In the previous post I introduced Dedekind cuts as a means of constructing the real number line, and I said that this perspective is responsible for the completeness of the real numbers $\mathbb{R}$.

Now, that was completeness in the topological sense.  There is another, very different notion of algebraic completeness.

A number system is called algebraically complete if every polynomial equation in one variable with coefficients from that number system can be solved in that number system.