Questions related to the algebraic structure of algebraic integers

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4
votes
3answers
851 views

Discriminant of a monic irreducible integer polynomial vs. discriminant of its splitting field

Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that ...
1
vote
0answers
118 views

Recovering unique factorization of elements by only looking at $p^n$-th powers?

I remember hearing somewhere that if $p^n$ is the largest power of a prime $p$ dividing the class number $h_K$ of a number field $K$, then there is unique factorization of the $p^n$-th powers of ...
3
votes
1answer
114 views

Congruences mod primes in Galois extensions

I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields ...
7
votes
3answers
2k views

Integral solutions to $y^{2}=x^{3}-1$

How to prove that the only integral solutions to the equation $$y^{2}=x^{3}-1$$ is $x=1, y=0$. I rewrote the equation as $y^{2}+1=x^{3}$ and then we can factorize $y^{2}+1$ as $$y^{2}+1 = (y+i) \cdot ...
2
votes
1answer
251 views

Ideals in a Quadratic Number Fields

In the literature it is stated that to each quadratic irrational $\gamma=\frac{P+\sqrt{D}}{Q}$ there is a corresponding ideal $I=[|Q|/\sigma , (P+\sqrt{D})/\sigma]$, where $\sigma=1$, if $\Delta ...
9
votes
1answer
494 views

What is the relationship between non-Archimedean places of infinite extensions of number fields and primes in the ring of integers?

Let $K$ be a number field and $L$ an infinite algebraic extension of $K$. Fix a non-trivial absolute value $v$ on $K$ (so $v$ is induced either by an embedding into the complex numbers or by a prime ...
7
votes
2answers
543 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
4
votes
3answers
596 views

Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$

I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as ...
46
votes
5answers
4k views

Are all algebraic integers with absolute value 1 roots of unity?

If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that αn=1 for some n). For example, (3/5 + 4/5 i) is not a root of ...
4
votes
2answers
605 views

Factoring prime ideals in a Galois extension of Q

Suppose $K$ is the splitting field of $x^3-2$ over $\mathbb{Q}$, and let $\mathcal{O}_K$ denote its ring of integers. I want to show that for any prime number $p$, $p\mathcal{O}_K$ is not a prime ...
12
votes
3answers
2k views

What are the units of cyclotomic integers?

This question made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many integers ...
2
votes
1answer
602 views

Algebraic Integers in a Cyclotomic Field

Let $p >5$ be a prime number. Prove that every algebraic integer of the $p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the ...
10
votes
1answer
980 views

Euler's remarkable prime-producing polynomial and quadratic UFDs

Good example of a polynomial which produces a finite number of primes is: $$x^{2}+x+41$$ which produces primes for every integer $ 0 \leq x \leq 39$. In a paper H. Stark proves the following result: ...
2
votes
2answers
209 views

Multiples of 4 as sum or difference of 2 squares

Is it true that for $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} - y^{2}$ for $x,y \in \mathbb{N} \cup (0)$. I was just working out and this came out to be true from $n=1$ to ...
9
votes
2answers
459 views

Primes dividing the values of integer polynomials

Problem: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$. The polynomial can be re-written as ...
8
votes
5answers
471 views

Applications of class number

There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it? It is nice if it is $1$; we have unique factorization of all ideals; but ...