Talk:Separable extension

Suggestions

shouldn't it be said anything about non-algebraic separable extensions?

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and we seem to be missing any mention of separability degree... Dmharvey 00:40, 18 April 2006 (UTC)

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I made a change in the Section "Separable and inseparable polynomials", namely it should be deg(f) pairwise distinct roots in some extension (just added pw. distinct to be more precise, but that's the point here). — Preceding unsigned comment added by 2A02:908:695:D280:5C35:810:58C0:4619 (talk) 06:24, 22 January 2018 (UTC)

Example of non-perfect field

A concrete example of one of these would be pretty cool to have in the article... -140.105.47.84 (talk) 07:24, 17 September 2009 (UTC)

Well, there is one (function fields). To specialise, take K(T), rational functions in T where K has characteristic p. This is purely inseparable over its subfield K(T^p). Which is isomorphic as field ... so the rational function field is not perfect. Charles Matthews (talk) 11:15, 17 September 2009 (UTC)

Major changes to article

I have re-written and expanded the article to cover the notions of a purely inseparable extension, and that of the separable part of an algebraic extension, and to provide a more comprehensive treatment of separable extensions in their own right. The article also includes an "Informal discussion", which provides at least some basic background for the article, and a section solely on separable polynomials (which includes an example of an inseparable polynomial as requested in the section above). Any comments, suggestions, or further improvements would be very much welcome. PST 08:48, 9 April 2010 (UTC)

Use of phrase "non-zero"

Why the repeated use of phrases like "non-zero prime" and "irreducible (non-zero) polynomial"? A prime element is by definition not zero. Similarly, a irreducible polynomial is by definition not constant, thus not zero. Not using the unnecessary "non-zero" phrase is exactly the "less is more" philosophy that is abstract algebra. Bender2k14 (talk) 00:54, 9 March 2011 (UTC)

I think you'd have to ask the editor User:Point-set topologist why s/he thinks it's necessary to say non-zero all those times (this is the editor that wrote all of these, see the diff [1] and all the following ones). Perhaps s/he was confused with prime elements versus prime ideals (the non-zero ones of the latter correspond to the former in the case at hand). Also, I'm pretty sure not saying unnecessary things is an idea that pervades much of mathematics. RobHar (talk) 04:33, 9 March 2011 (UTC)

You are absolutely right, RobHar. I will work to remove "non-zero" where unnecessary. Thank you for pointing this out! --PST 22:35, 29 September 2011 (UTC)

A paragraph in the lede

The following was moved from the lede

The study of separable extensions in their own right has far-reaching consequences. For instance, consider the result: "If E is a field with the property that every nonconstant polynomial with coefficients in E has a root in E, then E is algebraically closed." Despite its simplicity, it suggests a deeper conjecture: "If $E\supseteq F$ is an algebraic extension and if every nonconstant polynomial with coefficients in F has a root in E, is E algebraically closed?" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions; for instance, in the case corresponding to the extension $E\supseteq F$ being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.

I have heard of this matter. That, of course, doesn't mean it's not important. But I simply fail to see the significance. -- Taku (talk) 19:29, 14 September 2011 (UTC)

Taku: your two sentences "I have heard this matter. That, of course, doesn't mean it's not important" confuse me. Do you mean that "I have not heard this matter. That, of course, doesn't mean it's not important"?

I think the purpose of the paragraph you have quoted is to explain the importance of the theory of separable extensions. The theory of separable extensions is important in Galois theory and this is well established. However, there are other contexts in which this theory is important and one of them is explained in the paragraph you have quoted. --PST 22:39, 29 September 2011 (UTC)

(Yes, "not" was missing.) As I said, I simply cannot make sense of the paragraph. Especially, the part "deeper conjecture: ....". Separability is just some technical condition which has nothing to do with "algebraically closed." As I understand, the problem of "separability" comes up in algebraic geometry and the lede should elaborate on that. -- Taku (talk) 13:47, 30 September 2011 (UTC)

Also, it would be nice if you didn't just undo the edits, which simply invites the edit war. Finally, I created separable algebraic extension so that you can have "your stuff" in wikipedia. -- Taku (talk) 13:49, 30 September 2011 (UTC)

Taku, per Wikipedia:Consensus, it is your burden to explain your edits because you were the first one to revert a significant amount of content. You write: "Also, it would be nice if you didn't just undo the edits, which simply invites the edit war.". Is this not exactly what you did first?

As I said, I am not undoing the edits for the sake of it. The fundamental reason is that the theory of separable algebraic extensions is very important in mathematics and there is absolutely no mention given to it in the article as it stands. Indeed, this is the point of view expressed by virtually all major references on the subject. However, you are right that separable extensions that are not necessarily algebraic are also important. Therefore, I maintained the material that you have written. No, I do not wish to have "my stuff" on Wikipedia. I think you are misunderstanding the purpose of my edits.

If you delete material and if it is reverted, then it is your burden to explain the reasons for your deletion and it is up to the editors involved to reach a consensus. I encourage you to either reach such a consensus here or on Wikipedia:WikiProject Mathematics rather than simply reverting my edits. I am certainly not interested in engaging in an edit war but I am simply following the policies of Wikipedia:Consensus. Finally, I too must reach a consensus which is that which I am attempting to do now. --PST 00:35, 1 October 2011 (UTC)

I already explained myself. But I will simply repeat myself. The contents that were removed are already covered elsewhere like perfect field, separable polynomials. I don't disagree that separable algebraic extensions are important and that's covered. A connection to Galois theory is also mentioned in the lede. Finally, you didn't resolve the problem I raised: a cryptic paragraph in the lede. I don't have any problem seeing it in the lede if it makes sense. -- Taku (talk) 12:50, 1 October 2011 (UTC)

Taku: I have requested consensus at Wikipedia talk:WikiProject Mathematics. Let us wait for other editors to discuss the matter before we undo the edits of each other. Yes, some of the contents of article X are covered in many other articles different from article X but this is no reason to remove the material in article X. Some of the material in your revision of the article is contained in Kahler differential and Separable algebra but that is no reason to remove it and in any case it is not a wise idea to remove all redundancies from the entire Wikipedia network.

In this case, separable algebraic extensions are more fundamental to mathematics than separable extensions that are not necessarily algebraic in the same way that vector spaces are more fundamental to mathematics than Lie algebras. Both notions are important but one is clearly more ubiquituous than the other. Also, a single sentence explaining the connection to Galois theory is certainly not enough.

You write "I don't have any problem seeing it in the lede if it makes sense." Sure, it makes sense as it is mathematically correct. I do not understand your objections to this paragraph. You have also written that you fail to see its significance but as I explained to you, the lede should dicuss the importance of separable extensions in the general context of abstract algebra; Galois theory is one example but there are other examples as well and the paragraph that you have quoted is one such example. --PST 13:26, 1 October 2011 (UTC)

I have just seen this discussion, which is now old of eight months. Here is my opinion.

1. It seems to me that the contested paragraph is WP:OR. Not the results occurring in it, but the fact that they are a justification of the importance of the notion of separable extension. Thus this paragraph has to be removed.
2. It should be added in the lead that, in characteristic 0, all extensions are separable, and thus that the theory make sense only in positive characteristic.
3. IMO the importance of the notion of separable and inseparable extensions is that the existence of inseparable extensions is the main obstruction to extend many theorems from characteristic zero to positive characteristic. For example Hironaka's theorem that the resolution of singularities is possible in characteristic zero is not known in positive characteristic because of the existence of inseparable extensions. Thus its extension to positive characteristic requires a better understanding of inseparable extensions. I think that this example is much more convincing that the contested paragraph.
4. About Galois theory: the point is not that separability is useful in Galois theory. It is that the main result of Galois theory, the correspondence between subextensions and subgroups, is true in positive characteristic only for separable extensions. By the way, this is a more elementary example than the above one to show the importance of the subject.

D.Lazard (talk) 20:33, 11 June 2012 (UTC)

References

Old and new definition of separability

As discussed in http://en.wiki.x.io/wiki/Talk:Separable_polynomial there are two approaches to defining a separable polynomial, and the modern one is to require (deg f) distinct roots.

Currently the article claims equivalence of the following claims:

The constant 1 is a polynomial greatest common divisor of f and f'.[7] The formal derivative f' of f is not the zero polynomial.[8]

Note that in the source itself these are never said to be equivalent, the first is said to be equivalent to the modern definition of separability (which has a different name in the source), the second to the old definition (which the source calls separable).

These two claims are obviously not equivalent, and their equivalence is not actually supported by the source. Could the edit http://en.wiki.x.io/w/index.php?title=Separable_extension&diff=853049965&oldid=853019569 be undone? — Preceding unsigned comment added by 2A09:80C0:192:0:58C8:11AD:1DF:87ED (talk) 14:30, 24 October 2019 (UTC)

The article does not claim the equivalence of these assertions for general polynomials. It claims only this equivalence for irreducible polynomials and minimal polynomials (which are necessarily irreductible). Indeed this equivalence is true for irreducible polynomials. So, there is nothing wrong in the article. However, I'll edit the article for emphasizing that the equivalence applies only to irreducible polynomials. D.Lazard (talk) 15:32, 24 October 2019 (UTC)