Wikipedia:Reference desk/Archives/Mathematics/2010 October 19

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October 19

Quick query on field extension K-homomorphisms and isomorphisms

Hi all,

I'm looking at K-homomorphisms: L->L in the field extension L/K (i.e. homomorphisms: L->L which are the identity on the subfield K), and trying to show when they must be isomorphisms. I've shown that for any algebraic field it must be isomorphic, and in general any k-homomorphism must be injective no matter whether or not the field is algebraic (looking at the kernel of the homomorphism which is an ideal). However, I'm trying to find an example of a non-bijective K-homomorphism on a transcendental field, and I've come up with the following, which is not even injective so it must be wrong.

Look at K[X], and take the map F:K[X]->K[X], F(p(X))=p(1).

Then F(p(X)q(X))=p(1)q(1)=F(p(X))F(q(X)), F(p+q)=F(p)+F(q) and F(k)=k for any constant k, so F(1)=1 in particular and F is a K-homom. However, first question: F is clearly not injective (F(x-1)=0=F(0)), but what's gone wrong? I can see how it's really a map F:K[X]->K in a sense, but K is a subset of K[X] so presumably that's fine (or else there's no such thing as a map K->K which is not surjective). Which part of my argument is fallacious?

And my second question is (assuming it is my example rather than my logic which is wrong!) could anyone suggest such a non-bijective K-homom. which would work, over a non-algebraic field extension?

Thankyou very very much, just brief answers will suffice, I don't want to take up too much of your time! Spalton232 (talk) 00:08, 19 October 2010 (UTC)

K[X] is not a field. Your F doesn't extend to the field K(X), since F(X-1)=0, so there's nothing F(1/(X-1)) can be. The map from K(X) to K(X) given by F(X)=X² should be a non-surjective homomorphism. Algebraist 00:12, 19 October 2010 (UTC)

More complex number trigonometry

1. Simplify ${\frac {1+\cos 2x-i\sin 2x}{1-\cos 2x+i\sin 2x}}$

2. Show that ${\frac {1-\cos x+i\sin x}{1+\cos x+i\sin x}}=ie^{-ix}\tan {\frac {x}{2}}$

For the first one I have multiplied the numerator and denominator by a wide array of assorted rubbish (including the complex conjugate of 1-cos2x+isin2x), but I have no way of knowing if it's the simplest form. I wouldn't have a clue how to do the second one. --MrMahn (talk) 04:08, 19 October 2010 (UTC)

For the first one, "wide array of assorted rubbish" does not sound very systematic. I would notice that $1-\cos x+i\sin x=1-z$ for $z=e^{i2x}$. An narrow array of systematic rubbish to try would then be $1+z$, $1-z$, $1+{\bar {z}}$, and $1-{\bar {z}}$. One of these should end up with something you can express with a single term. –Henning Makholm (talk) 06:35, 19 October 2010 (UTC)

If everything else fails, the algorithm I proposed in response to your previous question should work for the second one, if you set $y=x/2$ and unfold the exponential using Euler's formula. Or (easier in practice) set $z=e^{ix/2}$ and rewrite the two sides as ${\frac {1-z^{-2}}{1+z^{2}}}=iz^{-2}{\frac {(z-z^{-1})/2i}{(z+z^{-1})/2}}$ after which only algebra is left. –Henning Makholm (talk) 06:21, 19 October 2010 (UTC)

(After ec)

1. Try this way:
   
   ${\frac {1+\cos 2x-i\sin 2x}{1-\cos 2x+i\sin 2x}}={\frac {1+\cos ^{2}x-\sin ^{2}x-2i\sin x\cos x}{1-\cos ^{2}x+\sin ^{2}x+2i\sin x\cos x}}$
   
   $={\frac {2\cos ^{2}x-2i\sin x\cos x}{2\sin ^{2}x+2i\sin x\cos x}}={\frac {\cos x(\cos x-i\sin x)}{\sin x(\sin x+i\cos x)}}$
2. Did you try to substitute $f(x)=f\left(2{\frac {x}{2}}\right)$ for sine and cosine of x? For the right-hand side use the Euler's formula.

HTH. --CiaPan (talk) 06:24, 19 October 2010 (UTC)

related to the question above

This is related to my question about refuting a famous open conjecture. Should the proof be titled "A disproof of" ___ conjecture or "A proof that" __ conjecture "is false"?

How should that be capitalized in the title? (title case or just initial capital only). 92.229.12.100 (talk) 11:40, 19 October 2010 (UTC)

I would go with the first option, shorter titles are more catchy. As for capitalization, it does not really matter. If you are going to publish it, the journal will impose their house style anyway.—Emil J. 12:24, 19 October 2010 (UTC)

Actually, in the second title the "A proof that" part is redundant. You can call it just "__ conjecture is false".—Emil J. 12:33, 19 October 2010 (UTC)

So would you use "The ___ is false" or "Disproof of ____" / "A disproof of _____" ? 92.229.12.100 (talk) 14:03, 19 October 2010 (UTC)

Here's some examples of titles people have used in such circumstances: A counterexample to Borsuk's conjecture (Borsuk's conjecture), Two complexes which are homeomorphic but combinatorially distinct (Hauptvermutung), Disproof of the Mertens conjecture (Mertens conjecture), On Hamiltonian circuits (Tait's conjecture), and A disproof of a conjecture of Pólya (Pólya conjecture). Algebraist 14:54, 19 October 2010 (UTC)

Personnally (perhaps because I'm a little older?), I dislike both alternatives. "A counterexample to" in my ears is much better. One reason is, that a conjecture wasn't stated as a theorem. When you write "A disproof", in my (older) ears it sounds a bit like "Well, this guy thought that (s)he had more or less proved that statement; but I'm going to show that (s)he was wrong". A properly stated conjecture is nothing like a loosely `claimed fact'. The person(s) putting it are not proven to have made any kind of mistake, if the conjectured statement is proved to be false.

If you provide a counterexample to a "famous open conjecture", and your example is a correct counterexample, and correctly motivated, then your own fame is ascertained anyhow. The professionals will go for the mathematical content, not for the title; and in fact a title which does not overstate the result probably is the better.

Actually, there are even more understated formulations, which sometimes are better, like "A negative answer to a question by...". It does happen that a mathematician thinks (s)he did not make a conjecture, just pose an open question; and in such cases may be a bit irritated even over a title like "A counterexample to a conjecture by...". (I'm not even trying to imagine the reaction to a title like "A disproof of..." in such a situation.) You probably should check the original formulation of the problem, before you decide of the title.

If the statement first was stated as a theorem, but later the proof was found insufficient, then "A disproof..." could be appropriate, I think. JoergenB (talk) 18:41, 19 October 2010 (UTC)

Rectification

What is the origin of the term Rectification? What is the connection between the original sense of the term (action of setting right) and the action of cutting of corners of a polyhedron? --İnfoCan (talk) 20:12, 19 October 2010 (UTC)

The original sense of the term is not setting right, but setting straight (it is derived from Latin rectus). This is where another geometric term, rectification of curves, comes from. I don't quite understand how (or whether) this is related to cutting polyhedrons.—Emil J. 14:43, 20 October 2010 (UTC)

The link he gave, Rectification (geometry), explains quite clearly how the word applies to cutting polyhedrons. Use of the word is quite old, like most solid geometry. I don't think it's likely we can discover exactly who first used it this way. But it makes sense to think that it is derived from the use in rectification of curves; the diagram for a rectified curve suggests cutting off. 68.36.117.147 (talk) 01:43, 22 October 2010 (UTC)

I see no such explanation in the article. Can you point to it more precisely? –Henning Makholm (talk) 02:41, 22 October 2010 (UTC)

how does arXiv work, or help with peer review?

Hi,

The arXiv article says "the arXiv is not peer-reviewed", so how does putting an article on arXiv before publication help / how do you use arXiv for peer review? If I put a paper up, will I just get emails from people who happen to be interested, and is that what it is good for? Or, should I myself email people who might be interested, linking my arXiv paper, and hopefully they will give me valuable feedback? Or, how is arXiv supposed to be uesd?

Basically what I'm saying is that I'm kind of fuzzy about the difference between me->arXiv->Journal versus me->Journal... thanks! 92.230.234.134 (talk) 20:29, 19 October 2010 (UTC)

It doesn't help with peer review. It won't help to get your paper published. It's just a big electronic library for people to put their papers in order to aid the dissemination of knowledge. More people will get to read your paper and that will make you work known to a wider base. Hopefully that will increase possible collaborations and citations. Both of which are good for a long and successful academic career. — Fly by Night (talk) 20:44, 19 October 2010 (UTC)

Some journals have integrated arXiv to their submission system. E.g. if you want to submit a new paper to one of the Physical Review journals, all you have to do is give your arXiv preprint number. Putting a paper on the arXiv may get feedback from other authors who will advertise their papers to you (asking for you to cite their paper). It helps making your paper more visible and will lead to your paper being cited a lot more. That in turn will help you getting your next paper published, because a Referee will, of course, check if the sort of research you are doing is of interest. Having published a few papers with zero citations doesn't help here. Count Iblis (talk) 03:51, 20 October 2010 (UTC)

Two reasons to post on the arxiv: 1) peer review can take a looong time, but if you post something on the arxiv it appears the next (working) day, 2) in contrast to most journals, the arxiv is accessible freely to all. (Virtually) no quality control is applied to arxiv postings - lots of stuff on there is wrong/cranky, but it's a great way to get your work out there. Tinfoilcat (talk) 18:43, 20 October 2010 (UTC)