What's a good book to attempt to learn this stuff, for future reference?

Gruson/Serganova "A Journey through Representation Theory" has lots of good stuff

Thanks!

"Let $K$ be the splitting field of $x^6+3$ over $\Bbb{Q}$." Does the fact that the definite article is used indicate that the sentence is talking about the smallest splitting field? Would that just be $\Bbb{Q}(r_1,...,r_6)$, where $r_i$ are roots of $f$?

yes

"the" splitting field is the smallest one

"Let $\omega$ be a primitive sixth root of unity over $\Bbb{Q}$." What exactly is $\omega$...I don't understand.

it's a generator of the group of sixth roots of unity over $\Bbb Q$

So, $e^{i \frac{\pi}{6}}$?

if you embed everything into $\Bbb C$, that's either $e^{2\pi i/6}$ or $e^{5 \cdot 2 \pi i/6}$

you're off by a factor of two, but otherwise, yeah, that's a possible choice

"over $\Bbb{Q}$"? What does that mean? I know what what it means to say that a vector space is over something.

well, it is contained in some field extension of $\Bbb Q$. There's also a primitive sixth root of unity in $\Bbb F_7$, for example

Since $7$ is prime, I can take $\Bbb{F}_7 = \Bbb{Z}/7$?

yes and the multiplicative group is cyclic

Ah, thanks!

Von Neumann correctly predicted how cells replicate before Watson and Crick did lol

sniped

What does this syntax bring to mind? $C_3^0$

Don't some people use that for binomial coefficients?

Well, it's meant to be a semigroup. Cyclic group of order 3 unioned with a "zero."

@Rithaniel i'd jokingly call it carbon but that's ${^{12}}C$

but i'd suspect that "C with sub/superscripts" is hopelessly overloaded in general

Fair. Though "$\mathbb{Z}/3\mathbb{Z}\bigcup\{ a\} $ where $ax=xa=a $ for all $x\in\mathbb{Z}/3\mathbb{Z} $" is a bit of a mouthful.

yeah

hmm, would a^2 be different than a? I mean, one has $a^2 x = a(ax)=a^2=xa^2$ as well

(if it's a zero, then I'd expect $a^2=a$)

Ah, yeah, that's an oversight in my expression of the semigroup.

It should be $ax=xa=a$ for all $x\in\mathbb{Z}/3\mathbb{Z}\bigcup\{ a\}$

Could also just do it as $ax=xa=a=a^2$ for all x in Z/3Z

That works too.

what i like about that example: 1 in Z/3Z still acts like an identity element for "most" of the semigroup

Well, it actually is an identity, still. I suppose I should refer to it as a monoid, and not a semigroup.

ah, yeah

the issue isn't the identity, it's the lack of an inverse for a

Yes, that prevents it from being a group.

More explicitly it is the multiplicative structure on the field of four elements, though.

neat

whats that theorem called listing a bunch of equivalent conditions for a riemannian metric to be complete? its not riemann roch

ok it was hopf-rinow

unfortunately doensn't contain the cases I was interested in :S

@Rithaniel I wonder about the inverse problem

Given such a structure, is it the additive structure of some ring

or field

You mean the multiplicative structure, right?

Considering the additive structure it's pretty boring.

Well, not boring, but pretty well-explored, at the very least.

hi DogAteMy, @Rithaniel

Heya Ted.

Currently trying to look up the statement of the fundamental theorem of finitely generated abelian groups.

That's not so difficult.

@s.harp what are you looking for

howdy @Ryan

Hi ted

@ÉricoMeloSilva Zhou gave a fantastic talk today

I wonder if the answer is manifolds with boundary.

Yeah, the only issue is that I've chosen to look it up in a book which lacks a table of contents. :P

Did you try an index instead?

Or is the "book" lacking both?

Just found it.

So yeah, given a particular structure, if it is the additive structure of some ring, then it is an abelian group. If it's finitely generated, then it's isomorphic to $(\oplus_{k=1}^n\mathbb{Z})\oplus F$ where $F$ is a finite abelian group. I don't know the case of an infinitely generated abelian group, though.

@ÉricoMeloSilva I stumped him with a question

Also, yeah, it's not actually a book. It's the class notes from Abstract Algebra. I just have them bound up in a binder because there's so much of them.

Ah, sometimes books are more convenient.

Indeed, such as for an index.

Is there a rap stack exchange where I can post lyrics

I love exchanges with pompous undergraduates like this.

I bet there's a reddit for that, Ultradark, but probably not a stack exchange. There might be a stack exchange specializing in music, though.

yeah I just found "Music fans stack exchange" I'll look on reddit also

I've been doing too much abstract stuff. I should get into more concrete calculations again.

Hey secret is in the chat there!

I love the typo that reads "hatred symbols are not vectors"

lol

I guess I shouldn't correct wrong answers when the questions are themselves messed up. Answerers shouldn't understand what's correct.

@Thorgott: I think that was a Freudian hatred.

Ah, the classification of infinitely generated abelian groups is an open problem. I wasn't expecting that.

I'm not sure I even would know how to begin writing down any sort of definitive list.

Heya @Karl. Here's someone who might know :P

Heh

lol

Looking to release my first album by next summer

hi Demonark

I looked at that question

1 view = 1 respect

Hey @Ted!

anyone here a terpischorean?

Gesundheit!

ugh I have logorrhea