Talk:Field extension

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I think I remember from math class that it's unknown whether every finite-degree extension is a Galois extension. If that's the case, it should be mentioned here. That always seemed like an amazingly basic open question, if it is one.

Q(³√2)/Q has finite degree, is normal, but it is not separable, so it is not a Galois extension. --ReiVaX 17:50, 15 July 2005 (UTC)

Umm, it's the other way around. Every algebraic extension of Q is separable; Q(³√2)/Q is not normal. -- EJ 20:27, 9 January 2006 (UTC)

The question you half-remembered from math class is this: is every finite group the Galois group of some Galois extension of Q? The answer is indeed unknown. This is the inverse Galois problem. AxelBoldt 07:15, 23 March 2006 (UTC)

[edit] merge with algebraic extension

I can't think of any reason why this article and algebraic extension should be separate articles. can you? -Lethe 13:02, Jun 28, 2004 (UTC)

I agree that the information on this page and on algebraic extension could be redistributed in a better way. I would also like to see an example of a galois extension which is not abelian, if a simple example exists. Owen Jones 10:42, 14 January 2006 (UTC)

Just found this: http://www.emba.uvm.edu/~sands/papers/stpopsub.pdf. See p. 7 where the author gives an example of an non-abelian galois extension. (I didn't check, if he is right ;-)) mathaxiom 23:38, 13 March 2006 (UTC)

Just a sidenote, perhaps soemone can clean up the symbols in this page? It's kind of distracting to read...

[edit] Restructering of the article

I am currently restructering the article in order to collect the important definitions, which were scattered in the article, in the Definition section. MathMartin 13:32, 17 April 2006 (UTC)

Generally looks good, but why did you remove ([1]) multiplicitivity of degrees? Dmharvey 18:49, 17 April 2006 (UTC)

Because it is already covered at Degree theorem and not directly relevant to the article. I added a link to the theorem in the See also section. MathMartin 19:38, 17 April 2006 (UTC)

Actually, I wonder if it wouldn't be better to move degree theorem to something like degree of a field extension, and to flesh out that article a bit, including a definition of degree (with the infinite case as well), and perhaps a proof of the multiplicitivity result, and some examples. Also, I don't remember "degree theorem" being a standard term for this result. What do you think? Dmharvey 20:07, 17 April 2006 (UTC)

Sounds good, I like the new title. But I cannot help you fleshing out the article as I only know about the finite case. MathMartin 21:36, 17 April 2006 (UTC)

OK, well that was probably more than I intended to write in one go :-) But now we have some overlap, which is not necessarily a bad thing, but perhaps we could focus the examples a bit better. Dmharvey 01:20, 18 April 2006 (UTC)

[edit] Is a field extension a special tensor product?

Of course not every tensor product is a field extension, but what about the converse? The multiplication of dimensions makes me suspect this, but it's been awhile since I "studied" this stuff.(but i did ask a teacher if it was a tensor product and he said yes, on the other hand, he didn't spend any time pondering his yes. Also, a different teacher denied that it was a tensor product.) If I'm right, some words on that and a link to tensor product would be valuable.Rich 09:03, 27 November 2006 (UTC)