Welcome to MathOverflow

Question

19

5

So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?

EDIT: Of course $\overline{\mathbb{Q}} \neq \mathbb{C}$. I don't know what I was thinking.

edited Jan 7 2010 at 14:57

Pete L. Clark
39.6k●3●102●225

asked Dec 13 2009 at 14:33

Andrew Homan
263●1●8

3

I bet you were not expecting the answer to be a theorem. It's one of the coolest little theorems in all of Galois theory. – Harry Gindi Dec 13 2009 at 14:58

4

Incidentally, C is not an algebraic closure of Q, since it contains transcendental elements like e (or more generally, because it has uncountable cardinality). – S. Carnahan♦ Dec 13 2009 at 16:16

Pete L. Clark · Accepted Answer · 2009-12-13 14:51:53Z UTC

34

Theorem (Artin-Schreier, 1927): Let K be an algebraically closed field and F a proper subfield of K with $[K:F] < \infty$. Then F is real-closed and $K = F(\sqrt{-1})$.

See e.g. Jacobson, Basic Algebra II, Theorem 11.14.

answered Dec 13 2009 at 14:51

Pete L. Clark
39.6k●3●102●225

2

Sure, you put LaTeX in your answer, but I answered first! =) – Harry Gindi Dec 13 2009 at 14:53

4

@Harry: You beat him by three and a half minutes only. That is about the time it would take Pete to type in his answer plus look up the exact reference in Jacobson's book. A clear case of independent responses if I ever saw one. – Harald Hanche-Olsen Dec 13 2009 at 16:24

1

I know that! I just typed it out as fast as possible because I knew the answer and so did everyone else. I wasn't claiming he copied it, I was claiming, "FIRST POST!!!" – Harry Gindi Dec 13 2009 at 16:41

24

Original reference, for the record: Emil Artin und Otto Schreier: Eine Kennzeichnung der reell abgeschlossenen Körper, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5 #1 (1927) 225–231 (doi:10.1007/BF02952522). Quoth Zentralblatt: “In Ergänzung der Untersuchungen über die algebraische Konstruktion reeller Körper der Verf. (ibidem 5 (1926), 85-99; F. d.M. 52) wird bewiesen, dass die reell abgeschlossenen Körper identisch sind mit den Körpern, die durch endliche Erweiterung algebraisch abgeschlossen werden können, ohne selbst algebraisch abgeschlossen zu sein.’ – Harald Hanche-Olsen Dec 13 2009 at 18:04

39

Am I the only person who finds this kind of competition to be an unhealthy expenditure of effort? – Boris Bukh Jan 7 2010 at 15:50

show 2 more comments

Harry Gindi · Answer 2 · 2009-12-13 15:00:45Z UTC

21

The Artin-Schreier theorem says that every algebraic closure of finite index has index 2, and it's the algebraic closure of a real-closed field.

Page 299 of Algebra by Serge Lang. Google Books Link to the page.

edited Dec 13 2009 at 15:00

answered Dec 13 2009 at 14:48

Harry Gindi
8,868●2●33●96

Examples of algebraic closures of finite index

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.