I sould try harder to keep things in the correct vats, the balance in my brain is out of whack;

@anon

@Kevin, I am always scattering from the main project i have to complete :)

@PedroTamaroff breathe first, pedro

@Irina It happens to me too. It has happened to every grad student I've talked to. So I guess it happens to everyone.

@Kevin, you are also in 1st year ?

@Irina this is my 3rd

@Kevin, how long takes PhD there ?

@Irina It always depends. 4 years is fast. 7 years is long. Everything in between is pretty average

@Kevin, generously reworded :)

we seems to be done with cubic calculator; now gotta rest;

I got a friends, it's nice to have common interests ;)

@KevinDriscoll 3 years is cheating.

@N3buchadnezzar hi $\varnothing$instein

@N3buchadnezzar I agree. Though I know of someone who did it here. Work 7 days a week all day.

@KevinDriscoll That is my life now, although I can take a day off when needed.

@N3buchadnezzar That is rough. If I'm working hard I get up to around 60 hours a week with research and TA duties. But most weeks I'm not that diligent

Yeah, it is a lot of work. Although I do have a lot of extracurricular activites as well. So it is more fun that it sounds. Although I am more outside my door than inside :p

@ted will you say hi?

I'll say hi if someone says hi to me :)

@KevinDriscoll Doing mathematics for the fun of it is... Suprisingly fun.

@Ted Hiiiiiiiiiiiii

I was just answering an email from one of our grad students who has been put in a totally awkward position by a professional who is abusing his position.

Hiiiii @Kevin and @Charles.

@Ted That's dispicable.

@TedShifrin i always say hi

Yeah, I'm really pissed off at this guy ... at a different institution in our state.

@Ted There is a professor here who I constantly avoid because he has a reputation for doing similar things

He's taken helicopter parenting to all new levels. And I hope he hears I'm bitching about him here :D

Prepare the bitchcopter

I do not know anything about the guy, but he's emailed at least two faculty and one graduate student trying to line up tutoring for his child at UGA. Of all people, a faculty member somewhere else should know how fed up with helicopter parents we get and how inappropriate it is to be one!!! GROWL^5.

@TedShifrin Heya.

Howdy @Pedro ... I thought you had purposely escaped once I showed up :D

LOL @N3

@TedShifrin Nah. Why would I do that ? =O

it's your new meme, @Pedro

Indeed. Gotta let the kids take care of themselves, or they'll never learn.

@TedShifrin Oh...?

I am trying to find the expectancy value of the area of a triangle inscribed in a circle of radius R. Hmmm

So one of my students decided to name our "obligatory sentence" for linear independence proofs the Declaration of Linear Independence. :) I thought that was hysterical ... and then I found out some mathematician wrote such a thing in 1988

I don't understand. Is this a parent or a professional in the university, or both?

Ah, that's a cool problem, @N3buchadnezzar

Both, but professor at a different university, @Pedro. But then trying to use his position to coerce a favor from a grad student at our school.

@TedShifrin Ah, that is what I was guessing.

I have a question.

I am livid ... I was already upset that he was bothering faculty here to do favors for him and his child, who should be adult enough to take care of him/herself.

RANTS and signs retirement papers.

What is your question, @Pedro?

@TedShifrin Well, I am reading the following property about algebraic integers.

Whenever $\alpha$ is an algebraic integer, then $\Bbb Q[\alpha]=\Bbb Q(\alpha)$.

Indeed. Quite related to the question someone was asking last night that @anon and I addressed.

But I am not sure where in the proof it is used that $\alpha$ is an algebraic integer.

Oh, no need for it to be an integer. Just an algebraic number.

@TedShifrin Ah, I see. Figured. So typo in Ireland and Rosen.

@Pedro memegenerator.net/instance/42357709

That's an unfortunate typo :(

But I have plenty of errors in my own books, despite being o so careful, so I know it happens all too easily.

@TedShifrin They also have non-standard nomenclature. Not that it bothers me, it is just curious.

Odd, since it's a relatively recent book.

@TedShifrin They call a subset of $\Bbb C$ a $\Bbb Q$-module if it is a finitely generated $\Bbb Q$-module.

Same for $\Bbb Z$.

well, that's probably a convention that others will make for convenience ...

@TedShifrin Ah, dunno.

Using those concepts they prove the algebraics are a field and the algebraic integers are a ring.

The proof is really cool.

Right. Yes, I always am tempted to teach that proof when I do field theory for undergraduates and do not prove the primitive element theorem.

@TedShifrin What is the "primitive element theorem"?

It actually came up in the solution of a tricky problem I wrote for our high school math competition a few years ago. Here you go: When is $\cos(2\pi/n)$ or $\sin(2\pi/n)$ a rational number?

Primitive element theorem says any finite algebraic extension is given by a single element.

@TedShifrin Ah.

Well.

The lemma they use is the following.

Let $M=[\gamma_1,\ldots,\gamma_r]$ be the $\Bbb Q$ (resp. $\Bbb Z$)-module generated by $\gamma_1,\ldots,\gamma_r\in\Bbb C$.

If $\alpha$ is such that $\alpha\gamma\in M$ whenever $\gamma\in M$ then $\alpha$ is algebraic (resp. an algebraic integer).

@Pedro: The actual question I posed was: For how many $n\in\Bbb N$ can we pack circles of radius $1$ tightly between concentric circles of radius $n$ and $n+2$?

@TedShifrin Ponders.

Edited. ... For your question, it's sort of interesting to try to bound the degree of $\alpha+\beta$ and $\alpha\beta$ when you know the degrees, or to find the minimal polynomial therefor if you know the minimal polynomials of $\alpha$ and $\beta$.

@TedShifrin I'd have to think about it.

About all of the above? :)

@TedShifrin The circles thing.

You hate it when I make you think :D

Yeah, it was one of two or three problems in a sequence of problems I wrote.

@TedShifrin Well, I need to look at a sum of arclengths.

really?

@TedShifrin You want to pack circles of radius one perfectly between $n,n+2$, right?

Yes ...

hi

Hi @Twink

@TedShifrin Well, or look at some angles that give rise to tangent lines.

Right.

@TedShifrin I will think about it.

Okey dokey. I'll check in later. :)

@TedShifrin Ted.

Is Lang's Algebra too advanced?

no

that's the book I use

hello guys

and it's easy

hoffman is more difficult

@Twink What is the full title of your book?

Linear Algebra

I have it in PDF

do you want it?

@Twink I am not talking about that one. I am talking about "Algebra".

is there is a way to transport all the account informatons and to another one and then delete it?

@mohamez You can ask that in meta.stackexchange.com

that's what I am doing right now thank you

is sharing ebooks illegal here in chat?

@mohamez It probably depends on your countries regulations. I don't know.

I wouldn't use MSE which is USA based for sharing ebooks that are not allowed to be shared.

I think is't not a problem here in my contry

$\epsilon$

$\in$

@mohamez $\in$

hi

Does 0.441725735696674 resemble any simple fraction or something involving $\pi$?

I did try Wolfram Alpha, but alas it did not bear any fruits.

hello @mick

@N3buchadnezzar Is that a rounded number or a truncated one?

Truncated

@N3buchadnezzar How many digits of accuracy?

Four so far, I am running some longer computations for higher accuracy now.

0.4417

Ah okay, then it'll be hard ot tell

if you can get 10+ digits of accuracy you can use the ISC, do you know about it?

@N3buchadnezzar

Yeah

It should be the expectency value of the area of a triangle inscribed in the unit circle

@N3buchadnezzar ISC suggests theres a number of things it oculd be involving $\sqrt{\pi}$ based on what you have

will ponder a bit about it

Thanks for the help!

@mohamez hi :)

@Irina hi

hi

euh yes

I have to go guys sorry

goodnight

check out my questions if you like :)

@Pedro: it's very dry, and not enough good exercises. It's tough even for grad students, but you might like it. Dummit and Foote is verbose, but has excellent exercises. I woukd recommend Jacobson over Lang.

@TedShifrin Well, I do have Jacobson's BAI.

Good :)

@TedShifrin Jacobson is also a baddie, though. =)

He's more concrete than lang. He also does decomposition of modules in terms of Euclidean alg, as I recall.

What is the charactieristic of $Z_4 X Z_6$? Is it just characteristic 4 multiplied by characteristic 6 = 24?

or is it just characteristic 6?

since the relations in Z_4 are redundant

@DonLarynx Don't guess.

Think.

Pick a usual element of $\Bbb Z_4\times \Bbb Z_6$.

It is of the form $(m,n)$.

Oh, Ted.

I guess characteristic refers not only to fields ...

@TedShifrin He probably means order...?

Let's find out ...

Could be the smallest $k$ s.t. $ka=0$ for all $a\in R$?

So pick an $(m, n) \in \mathbb{Z_4} X \mathbb{Z_6}$. For example $(3, 5)$. Then $k = 12$.

I don't know

Do you have a definition, @Don?

where is a good source to read about this?

all of my sources suck atm

A characteristic p is such that p(1) = 0.

@DonLarynx Note that $12={\rm lcm}(4,6)$

I'm not sure what "this" means ... But $p$ normally means prime. Are they talking about fields or general comm rings?

@PedroTamaroff What is a good source online to read about this?

@DonLarynx I don't know what "this" is.

characteristics of cartesian products of rings

@DonLarynx Here you have the definition of the characteristic of a ring

for $n*a$ they define it as $(n-1)a(a)$

so in $\mathbb{Z_6}$ we have $n = 6$ so we get $5a(a)$. I don't know how to make sense of this @PedroTamaroff

@DonLarynx Come again?

proofwiki.org/wiki/Definition:Power_of_Element

under monoid

@DonLarynx You're misreading things. They define exponentiation recursively.

In fact they have a typo in the monoid part. It says $x$ when it should say $a$.

Okay, so....What do they mean by: $n∗a$ they define it as $(n−1)a(a)$. Can you please give me an example?

@DonLarynx Do you know what it means to define something recursively?

Not how they defined it

@DonLarynx What do you mean?

Please show me an example where the equation $(n−1)a(a)$ is used

for $na$

@DonLarynx What they mean is this.

Consider the integers, for example.

Then, suppose you want to obtain $7\times 8$.

Here $n=7$, $a=8$.

Look at the integers additively, forget about multiplication.

Then to obtain $7\times 8$, you do $8+8$.

Then do $8+(8+8)$.

And so on.

Until you get $7\times 8$ which is defined here, as $8+8+8+\cdots+8$.

Seven times.

The recursion tells you how to obtain $na$ from $(n-1)a$.

Namely, $na=(n-1)a\cdot a$.

got it, thanks!

it makes so much sense now

so for $\mathbb{Z_3}$, we can say $((1 + 1) + 1) = 0$? @PedroTamaroff

@DonLarynx Yes.

We need more people like you in this world

@PedroTamaroff But the question now is different. First, do you agree the $lcm(a, b)$ is the characteristic of $\mathbb{Z_a} \times \mathbb{Z_b}$

@DonLarynx Use \times

I don't understand how though]

Can you point me in the right direction?

@DonLarynx Well, first, do you agree that if $n={\rm lcm}(a,b)$ we have $n(m,q)=0$ for any $(m,q)\in\Bbb Z_a\times\Bbb Z_b$?