Suppose that $S$ is a finite set with an odd number of elements. Then every involution $f:S\rightarrow S$ has a fixed point.


Application: Every prime of the form $p=4m+1$ may be written as a sum of two squares. The result above is used on p.20 [here][1].

Also, although not of the usual fixed point theorem form, is something I call the fixed point factor theorem. If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a polynomial and $n>1$, then $f(x)-x$ is a factor of $f^n (x)-x$ for natural $n$. This has a very obvious generalisation...


  [1]: http://www.iecn.u-nancy.fr/~chassain/djvu/Proofs-from-the-Book-2004.pdf