Talk:Transcendental extension

message: "The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma."
What is an "exchange lemma"? I'm unable to find other uses of this phrase - nonstandard terminology, perhaps? —The preceding unsigned comment was added by 70.48.179.234 (talk) 14:23, 13 May 2007 (UTC)
replies: text: I suppose it is an (incorrect?) translation of the German "Austauschlemma" ("Given a basis and an element v (not 0) of a vector space, one can replace one of the basis elements by v to get a new basis" -- see de:Basis (Vektorraum)). But I don't know of a corresponding theorem for the transcendence degree.--129.70.15.202 (talk) 19:33, 18 May 2008 (UTC)
replies: text: In matroid theory we have the "Basis exchange axiom" - which is the Austauschlemma of the previous writer. Matroid theory takes this Lemma as an axiom and builds on that. Perhaps a link to "Algebraic matroids" is in place here. Wandrer2 (talk) 10:00, 30 June 2008 (UTC)
replies: text: Milne has a lemma he calls the Exchange Property that says if β is algebraically dependent on {α1,...,αm} but not on {α1,...,αm-1} then αm is algebraically dependent on {α1,...,αm-1,β}.  That and the transitivity of algebraic dependence are all you need for the theorem about transcendence degree mentioned in the article. Chenxlee (talk) 17:56, 29 August 2008 (UTC)
replies: text:  The section also says "The dictionary matches algebraically independent sets with linearly independent sets;" ᛭ LokiClock (talk) 19:56, 24 May 2011 (UTC)
replies: text: It is currently linked to exchange lemma. 67.198.37.16 (talk) 19:26, 23 October 2020 (UTC)
thread_title:  "Exchange lemma" 

message: Trdeg is a short, new article that defines the notation trdeg and states a theorem relating the geometric idea of transcendence degree with the algebraic one.  I don't think this is specifically in this article already, but the last paragraph is certainly along the same lines.  Trdeg is just a stub and needs language cleanup and references; I think this article could use a reference but most abstract algebra texts should work. JackSchmidt (talk) 14:30, 21 November 2007 (UTC)
replies: text: I merged them, it seemed rather pointless having a separate article.  It'd be like having an article "Det" to describe the notation for determinants as well as the article on determinants itself. Chenxlee (talk) 20:18, 10 July 2008 (UTC)
thread_title:  Merge [[trdeg]] into this article 

message: It is not true that there are proper subfields of C isomorphic to C itself. It is actually a well known fact that C has only two field endomorphisms: the identity and the complex conjugation.
The reason is as follows: The complex numbers form a normal field extension over R, so any endomorphism of C restricts to an endomorphism of R.
However, the only endomorphism of R is the identity. Indeed, if f is an endomorphism of R, then f restricted to Q is the identity, since Q is the prime field of R. Furthermore, if x is a positive real number, i.e. the square of some non-zero real a, then f(x) = f(a^2) = f(a)^2 is a square of a non-zero real number, so f(x) is positive as well, and if x > y for any two real numbers, i.e. x - y > 0, then f(x) - f(y) = f(x-y) > 0, i.e. f(x) > f(y). So f is monotone. But any function on the reals that is monotone and preserves Q is the identity.
Now it follows for instance from Galois theory that there are exactly two endomorphisms of C extending the identity on R.
Please remove my post, it's incorrect.
— Preceding unsigned comment added by 129.132.146.160 (talk) 13:56, 27 July 2011 (UTC)
replies: text: The above paragraph is indeed incorrect. It's true that C is normal over R, but this does not imply that every endomorphism of C restricts to an endomorphism of R. By picking a transcendence basis S of C/Q and a non-surjective injection S -> S, we can construct a non-surjective field homomorphism Q(S)->Q(S) which extends to a non-surjective field homomorphism of the algebraic closure C -> C. AxelBoldt (talk) 23:35, 12 February 2023 (UTC)
thread_title:  Non-surjective endomorphisms of the complex numbers 

message: An editor has identified a potential problem with the redirect :Purely transcendental and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 June 10#Purely transcendental until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  1234qwer1234qwer4 10:46, 10 June 2022 (UTC)
thread_title:  "Purely transcendental" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]