# Examples of algebraic closures of finite index

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.

• 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. Dec 13, 2009 at 14:58
• Incidentally, C is not an algebraic closure of Q, since it contains transcendental elements like e (or more generally, because it has uncountable cardinality). Dec 13, 2009 at 16:16

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.

• Sure, you put LaTeX in your answer, but I answered first! =) Dec 13, 2009 at 14:53
• @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. Dec 13, 2009 at 16:24
• 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.’ Dec 13, 2009 at 18:04
• Am I the only person who finds this kind of competition to be an unhealthy expenditure of effort? Jan 7, 2010 at 15:50
• @Boris: You are not. Mar 22, 2011 at 20:03

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.