Talk:Algebraically closed field

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Algebraic numbers are algebraically closed?

I don't see how the algebraic numbers are algebraically closed. The definition I know of the algebraic numbers is that they are the set of real solutions to polynomials over the integers. Therefore 1 is an algebraic number. However has no solution in the algebraic numbers.

Is my definition of algebraic number wrong?

—Preceding unsigned comment added by Luqui (talkcontribs) 08:59, 14 December 2005

Definitively: Algebraic numbers are all the complex(!) solutions of polynomials with rational(!) coefficients. —Preceding unsigned comment added by (talk) 11:46, 7 November 2007 (UTC)

I did a bit of googling, and apparently some definitions of an algebraic nunber require it to be real, and some do not. I do not know which is more standard. - Jwwalker 23:30, 31 December 2005 (UTC)

this is probabely mixing two different things:
  1. definition: let L be a ring, K be a subset of L, then l in L is algebraic over K when: there is a polynomial p with coefficients in K s.t. p(l)=0. in some cases K is required to be a subring or something not just a subset.
  2. definition: let K be a field then K is algebraically closed when all polynomials with deg>=1 have at least one root (as defined in the article).

probabely the real algebraic numbers are: the Real numbers that are algebraic over Z.
the general are the algebraic closure of Z, which is C. --itaj 00:49, 14 November 2006 (UTC)
The algebraic closure of Z is not C. The algebraic numbers are the complex numbers that are algebraic over Q. These form a countable subfield of C. The real algebraic numbers are the algebraic numbers that lie in R. --Zundark 10:00, 14 November 2006 (UTC)
ahh, yes ofcourse, i'm sorry. the algebraic closure of R is C, and the closure of Z is a subfield of C. --itaj 16:08, 24 November 2006 (UTC)
I changed "algebraic number" to "(complex) algebraic number". Hopefully this does not create more confusion than it solves, but I certainly had the same reaction of "how can the (real) algebraic numbers be algebraicly closed?". The one book I looked up (Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics) does define algebraic numbers as complex, so if this whole concept of "real algebraic number" was just a misconception of mine, I'm OK with a revert. Kingdon 19:54, 22 March 2007 (UTC)

What field is this article in?

Is this really supposed to be number theory rather than algebra? I'm relucant to just change it in case I missed something subtle, but at first glance Algebraically closed fields is very much an algebraic topic. Kingdon 16:24, 13 May 2007 (UTC)

I've changed it to algebra. --Zundark 16:56, 13 May 2007 (UTC)

Some suggestions for improvement

This is a good start. Here are some further ideas:

1) Spell out more explicitly what it means for a polynomial to have a root: could it hurt?

2) The fact that x^2+1 = 0 has no root in the real numbers shows moreover that no subfield of the real numbers is algebraially closed. This would be a good opportunity to link to formally real fields, in which -1 is not a sum of squares -- they cannot be algebraically closed either.

3) Other nonalgebraically closed fields: pseudo-real closed fields, p-adic fields, formally p-adic fields, Laurent series fields, pseudo-p-adically closed fields, fields which are finitely generated over their prime subfields or over a field of rational functions in any number of indeterminates over any field, any field which is elementarily equivalent to a non-algebraically closed field, non-perfect fields.

4) Other algebraically closed fields: for any field k, the field of all elements which are algebraic over k is algebraically closed. (With k = Q, this recovers the algebraic numbers.)

5) Another characterization of algebraically closed fields: fields which are perfect and have trivial absolute Galois group.

6) References for the equivalent properties of algebraic closure?

7) Under "other properties", mention the maximal solvable extension of Q as a field which satisfies the given conditions.

8) Define algebraic closure before asserting that every field has one. (A better definition is an algebraic field extension which is algebraically closed, as is given in the link.) Give examples of algebraic closure of Q and R.

9) The statement "every field has a unique algebraic closure" is subtly inaccurate. There are many algebraic closures of a given field. What is true is that any two of them are isomorphic as F-algebras. However, there are many isomorphisms, one for each element of the absolute Galois group of F, which if F was neither algebraically closed nor real closed to begin with, is an uncountably infinite group. 11:33, 17 September 2007 (UTC)Plclark