About that field isomorphism… February 15, 2009Posted by Ben Webster in Galois theory.
There was a question about the isomorphism of fields between and . I just stuck that in this previous post as a comment, so let me elaborate a tiny bit.
The important thing about this kind of field isomorphism is that it tells you almost nothing interesting. What makes interesting is its topology, and when you lose that, becomes a rather floppy, uninteresting object.
So, I guess, my point is, this result is helpful as a security blanket, when you have to face the -adics the first time, but it actually has few really important consequences.
Theorem. Any two algebraically closed fields of the same transcendence degree over are isomorphic. In fact, any of isomorphism of subfields (EDIT: of finite transcendence degree over ) of and can be extended to a global isomorphism. In particular, if have the same minimal polynomial, then there is an isomorphism sending to .
Corollary. The transcendence degree of both and over is the continuum, so they are isomorphic.
I’ll just note, the moral here is that the isomorphism with tells you nothing about an element of you couldn’t have found out from its minimal polynomial. I’ll include a proof, just so you see how much non-canonical stuff goes into this isomorphism.
Proof. Pick any isomorphism between subfields of . Then pick transcendence bases for over . Since these have the same cardinality, there is a bijection between them, and thus a field isomorphism . Since these are transcendence bases, taking algebraic closure, we obtain an isomorphism .
EDIT: A better proof was suggested in comments, which I have stolen. Thanks, Lior.