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)

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)