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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). —-
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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 <-:
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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
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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
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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
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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
1053:. 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”.
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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.
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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
231:
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.
181:"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?
204:
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
1072:, 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
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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.
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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.) —
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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
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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
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Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a
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needs language cleanup and references; I think this article could use a reference but most abstract algebra texts should work.
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is clearly an excellent starting point, and this fully justifies the move request. If the move is performed, it will leave
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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
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The section also says "The dictionary matches algebraically independent sets with linearly independent sets;" á›
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is the identity. Now it follows for instance from Galois theory that there are exactly two endomorphisms of
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It wpuld be nice to have a comment on the relation between transcendence degree and vector space dimension
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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transcendental extensions, just as understanding peace is not equivalent to understanding wars. —-
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after discussing it on the closer's talk page. No further edits should be made to this discussion.
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on
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per those above. The current title sounds like an academic degree in transcendence.
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more closely related to trdeg. Regarding "subtopic", I feel that to cover trdeg in
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A transcendental extension is just one that is not algebraic, and we already have
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The reason is as follows: The complex numbers form a normal field extension over
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628:) which extends to a non-surjective field homomorphism of the algebraic closure
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209:). But I don't know of a corresponding theorem for the transcendence degree.--
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has only two field endomorphisms: the identity and the complex conjugation.
786:. I think this is much like the "flammable" and "non-flammable" example in
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is monotone. But any function on the reals that is monotone and preserves
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Knowledge:Redirects for discussion/Log/2022 June 10#Purely transcendental
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are specific to this theory. Transcendental extensions have a role in
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is a positive real number, i.e. the square of some non-zero real
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An editor has identified a potential problem with the redirect
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Milne has a lemma he calls the
Exchange Property that says if
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and rewrite it so it discusses such an extension in general.
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863:" I have in mind to justify keeping trdeg separate from
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The above paragraph is indeed incorrect. It's true that
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112:, a collaborative effort to improve the coverage of
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Non-surjective endomorphisms of the complex numbers
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831:Putting aside whether we need a standalone
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552:Please remove my post, it's incorrect.
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759:) 06:06, 18 March 2023 (UTC)
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120:and see a list of open tasks.
1147:The person who loves reading
1002:a trans field extension. —-
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