Good afternoon /sci/! This is your chance to show that you're

Undergrad fag here. What does (p, a)^(n)=(p) and that little cross symbol mean in the first question? And why algebraic number theory? Why not analytic? Or are you doing both? And aren't you in your first/second year?

PhD here. What is a "number"?

>>7767643
>what is the little cross symbol
gtfo

File: 1450704648920.jpg (6KB, 129x119px) Image search: [Google]

6KB, 129x119px

>>7767609
ho my christ, 2016 and still doing seriously classical algebra. I bet you paid for this.

when will leave ZFC ? when will you grow ?

>>7767609
I'm a math PhD student and I don't give a shit about number theory. Sorry, I'm not autistic.

>Let
[math]f = \sum_{k=0}^n a_{k} X^{n-k} [/math] with [math]a_0=1[/math]

I hate that shit.

>>7767643
>(p, a)^n=(p)
Those generated ideals are the same.
>that little cross symbol
It's the doesn't divide symbol as in p^2 does not divide a_n. The exercise itself is a trivial consequence of Eisenstein's criterion. A joke for everyone who took any class in Algebra.

>>7767609
>For this exam the use of lecture notes...is allowed
>Results in the lecture notes...may be used

Dropped.

>use separate sheets for the two problems
>writes directly on the exam paper
Can't follow instructions, automatic fail.

>>7767719
It's not that trivial anon. Why doesn't p divide [O_K:Z(alpha)]?

>>7767719

The number of problem sets I turned in that primarily consisted of the word "Eisenstein" and nothing else was hilarious.

> mfw full points
> O

>>7767609
>This is your chance to show that you're not all undergrad faggots...
That's funny, because I learned this in undergrad.

>>7767860

Trufax.

>>7767609
>majoring in mathematics
HAHAHAHAHAHAHA

>>7767860
Post proof faggot.

>>7768771
You've never heard of undergrads learning algebraic number theory? I took a fourth year course on it that went through the relevant sections of Cassels/Frohlich that cover the material on that problem set

>>7767851
How retarded are you that your professors test you multiple times extensively on the same concept?

>>7768771
How do I prove when I learned algebraic number theory?

>>7769052
>says he can do higher level math
>cant do simple proof

The only hard math thread at /sci/ at this moment, I am a bit underwhelmed.

>>7768789
I am an undergrad too (inb4 brag thread), but you kinda implied your uni did this as an undergrad course.

>>7769052
Why don't you solve 1(i) then?

Okay so I guess I'm retarded but how do I prove [math](p) \subset (p,\alpha)^n[/math] I have thought about it but I still don't see how to use the hypothesis.

>>7767737
At a certain point, professors stop making you need to know everything without any references because that's not the point of education. Groveling for gold stars is for babies.

>>7769669
You might want to use Kummer-Dedekind, but you have to prove it can be applied.

>>7767609
>final exam
>less than 50% of the final grade of the course

lmao American grade inflation.

>>7769693
>lecture notes are allowed

>>7769686
Oh okay, I did not know about this theorem. It changes everything !
I'll try to write a solution for the whole thing at some point today

>>7769694
I'm sorry that you are only used to 'write down the definition of X'-tier exams. There are a lot of ugly algorithms in this course and it seems quite pointless to me to try to learn them by heart.

>>7769698
Nice.

>>7769694
Who cares if lecture notes are allowed.
An actually challenging problem is not a recitation and should not become trivial if you have your notes with you.
If it is the case for your exams I would suggest considering transfering

>>7769669
Let [math]b_i = \frac{a_i}{p} [/math]
[eqn] \alpha^n = - a_1 \alpha^{n-1} - \ldots - a_{n-1} \alpha - a_n = p \left( -b_1 \alpha^{n-1} - \ldots - b_n \right)[/eqn]
[eqn] (p, \alpha)^n = (p^n , p^{n-1} \alpha , \ldots , p \alpha^{n-1} , \alpha^n) = (p^n , p^{n-1} \alpha , \ldots , p \alpha^{n-1} , p \left( -b_1 \alpha^{n-1} - \ldots - b_n \right)) = p (p^{n-1} , p^{n-2} \alpha , \ldots , \alpha^{n-1} , \left( -b_1 \alpha^{n-1} - \ldots - b_n \right)) = (p)[/eqn]

>>7769718
Why does the last equality hold?

>>7769693
This is Dutch, and besides more than half surely won't pass, and I don't think anyone got higher than an 8/10. So I don't understand what's inflated here.

>>7769760
From Eisenstein we know that the polynomial [math]f [/math] is irreducible.
Since [math] \alpha [/math] is the root of an irrudicible polynomial of degree [math] n [/math] we know that the only rational numbers [math] \lambda_0, \lambda_1, \ldots , \lambda_{n-1} [/math] with
[math] \sum_{k=0}^{n-1} \lambda_k \alpha^k = 0[/math]
are [math] \lambda_0 = \ldots = \lambda_{n-1} = 0 [/math].

Just think about what this implies because I have no fucking idea. The only algebra classes I ever took were linear algebra and an introduction to group theory.

>>7767609
>haha, yeah, I'll show all those internet strangers how cool I am!

>>7769807
Can't you stand the sight of actual, non calc1-tier math, anon?

>>7769799
>The only algebra classes I ever took were linear algebra and an introduction to group theory.

It's pretty good that you remembered Eisenstein's Criterion then.