Definition talk:Algebraic

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Ported 10-Mar-2013 from the deleted page "Algebraic"

we already have Definition:Transcendental which includes "algebraic", but in a slightly different context - you may want to merge the pages. --prime mover (talk) 23:06, 18 December 2008 (UTC)

Yeah, perhaps. Aside (again), it would be good if you didn't adopt the notation from your reference as "convention", since a lot of the time it has been rather odd. I don't know how to "merge" pages... I'll leave that to you. I'm hoping to get a result or two in about algebraic/transcendental extensions in before the night is out. :P --Grambottle 23:22, 18 December 2008 (UTC)

Which particular aspect of what particular notation is "odd"? This page (Transcendental) comes from a different source: Thomas Whitelaw: An Introduction to Abstract Algebra (Blackie, 1978). I thought his notation was pretty conventional. Now I don't know what the pukka convention is from baked beans, I couldn't care less what the convention is quite frankly, it doesn't matter squat because convention can be shoved where the sun don't shine. If a notation is convenient and does the job and is neat and easily rendered and isn't (too) ambiguous, it does for me. I've come to serious blows before with people who give me a hard time over the fact that I don't make the proper marks on the paper (I hospitalised somebody once who "refused to allow me" to use a particular mnemonic "because it's not the right mnemonic") because IMO it doesn't matter what convention you use as long as you explain what it means at the time. It's about as important as what end you open your boiled egg, i.e. just about twice as important as whether you're a xtian or a m'lim. (Perhaps this is why I've been banned from teaching mathematics.) --prime mover (talk) 06:31, 19 December 2008 (UTC)

I wasn't talking about your conventions with transcendental/algebraic - shoulda been specific. In fact, I like that you explained that in a slightly more general context. What I was mentioning was the ring of polynomial "forms" (not a disagreeable name) but that you then go to say convention is to use a capital X, which isn't the case. If you don't care what convention is, you shouldn't be claiming convention; instead, perhaps saying "we can write".

I think keeping up with convention is a very important thing on a site like this. Yes, there are authors with differences in notation. For example, Group Actions: using either $\cdot$ or $*$ or simply placing elements next to each other are all modern conventions; using a wedge product is not. I mean not to criticize negatively (don't think I have been) but it's important to be fussy about these things, especially if we want to attract more people to the site. Furthermore, if we want ProofWiki to be a reliable and readable website for people to use, we're going to have to cater to convention.

There ARE conventions I really don't like; one of which being the excessive use of "1", when dealing with groups, or writing abelian groups additively using "0" for the identity. However, such is modern convention, and I'd use that on this site despite my own preference.

For you, I really think it's a matter of picking better (and perhaps more recent) references. If you've got an academic library nearby, I strongly recommend that Rotman book. One of the best references I've seen so far. I need sleep - hope this clarifies my points. --Grambottle 09:03, 19 December 2008 (UTC)

Sorry I think all the above is utter bollocks. --prime mover (talk) 21:06, 19 December 2008 (UTC)

Well that's a shame. I do hope you come to understand the reason behind my statement eventually. I like most of your work but honestly some of the notation is simply out of date or exclusive to your sources. --Grambottle 21:13, 19 December 2008 (UTC)

I have to agree with Grambottle on this one. We need to pick some notation to use, and if there is a well established notation, then we should probably use it. That said, I do agree that there are bad conventions out there, but I think that we should stick to convention when we can, ard make a note of notations that are coming in to use, not the other way around. After all, if we aren't accessible to the average mathematician, then we certainly aren't doing our job right. --Cynic-----(talk) 21:27, 19 December 2008 (UTC)

"For you, I really think it's a matter of picking better (and perhaps more recent) references." Well, like as before I have proved that I'm never ever going to be a mathematician because I'm ignorant of the conventions. Ah bollocks to it, I can't be bothered. --prime mover (talk) 22:27, 19 December 2008 (UTC)

Sorry, I was tired and emotional last night.

So you say: "Group Actions: using either $\cdot$ or $*$ or simply placing elements next to each other are all modern conventions; using a wedge product is not." See Definition talk:Group Action. On the page itself I pointed out that there appears to be no consistent convention for this. You said it yourself, some sources use dot and some use asterisk. No convention exists. QEFD.

Okay yes, $F[X]$ as opposed to $F[\alpha]$. Oh dear I am a silly boy. My fault for taking seriously what I read in (and quoted from) my "Whitelaw: An Introduction to Abstract Algebra (1978)": "It is conventional to use $X$ to denote an element transcendental over a given integral domain" (Section 64.4). It was new when I did my degree. If the convention nowadays is to use $\alpha$ instead, then change the page to say so.

I have no academic library nearby (not all of us are lucky blighters to have unrestricted access to such) so you got to make do with what you can get. --prime mover (talk) 08:34, 20 December 2008 (UTC)

About the Group Actions: yes, notation varies, but typically never using the wedge product - again, it's a very special symbol, reserved for creating the Exterior Algebra of a Tensor Algebra and logical operations. Its use in algebra is already well-defined and conventional in the strongest sense (by this I mean things such as $\Z$ denoting the integers). The asterix, dot, or juxtaposition should suffice - these are the variants of common convention.

You're also missing the meaning of the proof I started the other night. See, when you adjoin an algebraic element to a field, you end up with the same field as the polynomial ring evaluated at $\alpha$; this is NOT the case with a transcendental element. The proof will contain a simple two-three line argument showing that the polynomial rings in a transcendental $\alpha$ and $x$ are, in fact, isomorphic; but such rings are NOT isomorphic to the field of quotients of that ring (again, that's actually TRUE in the algebraic case).

In the proofs to come in my development of Galois Theory, these kinds of rings will crop up ALL the time, for it is very useful to be able to express $F(\alpha)$ as a simple polynomial ring, with a basis of $1,~\alpha,\alpha^2,...,\alpha^{n-1}$ (where $n$ would be the degree of $\alpha$ over $F$.

I hope that clarifies your confusion as to why I would use $F[\alpha]$. --Grambottle 16:11, 20 December 2008 (UTC)

I've already got a proof in here somewhere that"the polynomial rings in a transcendental $\alpha$ and $x$ are, in fact, isomorphic" and I think there's a start of another one where "such rings are NOT isomorphic to the field of quotients of that ring" ... sorry to be such a doofus but this is all stuff I've had to teach myself.

And do what you wanna do with that Group Action stuff, just don't expect me to go and change everything I've done merely because of notation conventions. As I say, it makes me angry (my problem, I think I'm ill). --prime mover (talk) 18:58, 20 December 2008 (UTC)