Good afternoon /sci/!
This is your chance to show that you're not all undergrad faggots...
Time for Kummer-Dedekind and chill.
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
>>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
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
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.