This is a *proof* that a compact field is finite, so you might want to peek at that node to see what it's all about...

Let F be a compact field, and suppose to the contrary that F is infinite. By Robinson's "overspill" lemma, F (pseudo-) contains non-standard elements. Let y be in F^{*}\F (i.e. y is non-standard). Since F is compact, every pseudo-element has a standard approximation (see the non-standard definition of compactness in my writeup on "compact"). Let x be the standard approximation of y, and take z=y-x. Then z is non-standard (or y=z+x would be standard), and 0 is the standard approximation of z. In particular, z≠0.

Since "∀z.z≠0->∃w.w*z=1", we have an inverse w of our z. And since F is compact, w too has a standard approximation v. By continuity, 0*v=1. But this cannot be! (For instance, we deduce that 1=0*v=(0+0)*v=2*(0*v)=2, which is false).

So our assumption that F is infinite cannot hold; a compact field F must be finite.