Talk:Transcendental extension

Untitled

It wpuld be nice to have a comment on the relation between transcendence degree and vector space dimension

"Exchange lemma"

"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).

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)

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)

Milne has a lemma he calls the Exchange Property that says if β is algebraically dependent on {α₁,...,α_m} but not on {α₁,...,α_m-1} then α_m is algebraically dependent on {α₁,...,α_m-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)

The section also says "The dictionary matches algebraically independent sets with linearly independent sets;" ᛭ LokiClock (talk) 19:56, 24 May 2011 (UTC)

It is currently linked to exchange lemma. 67.198.37.16 (talk) 19:26, 23 October 2020 (UTC)

Merge trdeg into this article

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)

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)

Non-surjective endomorphisms of the complex numbers

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)

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)

"Purely transcendental" listed at Redirects for discussion

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)

Requested move 18 March 2023

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: MOVED. Hadal (talk) 18:15, 4 April 2023 (UTC)

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Transcendence degree → Transcendental extension – The link transcendental extension is currently a redirect while we have an article on a trans degree. This seems weird to me. So I suggest we move this article to move/override a transcendental extension and rewrite it so it discusses such an extension in general. Taku (talk) 06:06, 18 March 2023 (UTC) — Relisting. Favonian (talk) 09:10, 25 March 2023 (UTC)

A transcendental extension is just one that is not algebraic, and we already have algebraic extension. I think this is much like the "flammable" and "non-flammable" example in WP:OVERLAP so perhaps transcendental extension should not get its own article. The current title in some sense reminds us to avoid this overlap. (It seems that only itwiki has a standalone article for d:Q3733413, though of course this does not forbid us to have one.) —HTinC23 (talk) 18:04, 18 March 2023 (UTC)

No, that doesn’t work: algebraic extension does not discuss a transcendental extension (since the latter is not algebraic). There are many things unique to transcendental extensions such as transcendence degree. So, it does make sense to have an article on the topic. Understanding algebraic extensions is not equivalent to understanding transcendental extensions, just as understanding peace is not equivalent to understanding wars. —- Taku (talk) 18:21, 18 March 2023 (UTC)

Putting aside whether we need a standalone transcendental extension, another problem is that transcendence degree is not a subtopic of transcendental extension (trdeg > 0), but instead a subtopic of field extension (trdeg ≥ 0), so the title transcendence degree should not be turned to a redirect to transcendental extension. And since we are not discussing a merger to field extension, the article could stay here.

(The "WP:OTHERSTUFF" I have in mind to justify keeping trdeg separate from transcendental extension or field extension is dimension (vector space), or dimension of an algebraic variety more closely related to trdeg. Regarding "subtopic", I feel that to cover trdeg in non-algebraic extension is the same as covering divisor in non-(prime-or-unit) number or ideal class group in non-PID Dedekind domain or any X in non-Z Y where all Y's have X but a non-trivial X iff the Y is not Z. It seems weirder to me than the current situation. I apologize for resorting to bad analogies, running out of coherent arguments.) —HTinC23 (talk) 23:05, 18 March 2023 (UTC)

Of course, the transcendence degree is an invariant of a field extension. But the only interesting case is when the extension is transcendental. The same for transcendence basis. So, I think we can regard them as subtopics of transcendental extensions. For analogies, a trans.deg is an analog of a vector space dimension. The current setup, as I see, is like there is a dimension article but no vector space article, which is weird. By the way, the reason I suggested a move is also because the current trans.deg article contains quite a bit of materials on trans. extensions; e.g., purely transcendental extension. Such materials would feel more natural and can develop better if the article is named a trans extension (e.g., having a section on purely trans. extensions). —- Taku (talk) 06:44, 19 March 2023 (UTC)

Note: WikiProject Mathematics has been notified of this discussion. Favonian (talk) 09:11, 25 March 2023 (UTC)

The notion of transcendence degree (over the prime field) is also used as a dimension notion of a field itself, for instance in algebraic model theory. Therefore I think covering it separately from the notion of an algebraic/transcendental field extension is useful and would support it remaining at its current title. Felix QW (talk) 09:37, 25 March 2023 (UTC)

Regarding the analogy with vector spaces further, we do have an article on field extensions, so to me not having an article on transcendental extensions is more like not having an article specifically on vector spaces of positive dimension. Felix QW (talk) 09:38, 25 March 2023 (UTC)

This analogy doesn’t quite work since the zero-dimensional vec sp is trivial; there is nothing to study, while a field ext of trans degree zero (i.e., an alg field extension) is not trivial; there are a lot of stuff on them (e.g., Galois theory). Hence, the right analogy should be a vec space <-> a trans field extension. —- Taku (talk) 10:20, 25 March 2023 (UTC)

• Comment. There are two different implicit questions in the above discussion: (1) Should we have an article Transcendental extension? (2) If Transcendental extension becomes a stand-alone article, should Transcendence degree be a redirect to this article or to Field extension? If the answer to (1) is “yes”, Transcendence degree is clearly an excellent starting point, and this fully justifies the move request. If the move is performed, it will leave Transcendence degree as a redirect to Transcendental extension. It is only after the move that the target of could be reasonably discussed. Personally, I am in favor to not change the redirect left by the move, since “transcendence degree zero extension” is rarely used, as people prefer to use “algebraic extension”. D.Lazard (talk) 10:51, 25 March 2023 (UTC)
• Strongly support, per my above comment and per the following. There a rich theory of transcendental extensions that is far to be reduced to “a transcendental equation is a non-algebraic extension”. The concepts of transcendence basis and purely transcendental extension are specific to this theory. Transcendental extensions have a role in algebraic geometry that can be compared to the role of algebraic extensions in number theory. In particular, the transcendence degree is used for one of the definitions of the dimension of an algebraic variety. Lüroth’s theorem is a theorem on subextensions of a purely transcendental extension of degree one. Much work has been devoted for extending this theorem to transcendence degree two. These are example of content that must be added to an article Transcendental extension. D.Lazard (talk) 10:51, 25 March 2023 (UTC)
• Support per those above. The current title sounds like an academic degree in transcendence. BD2412 T 00:12, 3 April 2023 (UTC)
• Support per nom. The person who loves reading (talk) 14:58, 4 April 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.