Questions related to the algebraic structure of algebraic integers

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11
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1answer
252 views

If $u=\frac{1+\sqrt5}{2}$, then $u^3=2+\sqrt5$, but $u^2=\frac{3+\sqrt5}{2}$. What is the group that measures the power that makes units look nice?

For $A=\mathbb{Z}[x]/(f)$ with quotient field $K$ and ring of integers $B$, does $U(B)/U(A)$ have a name? For instance $u = \tfrac{1+\sqrt{5}}{2}$ is a unit in $\mathbb{Q}[\sqrt{5}]$, but neither ...
11
votes
1answer
150 views

If $p\equiv 1,9 \pmod{20}$ is a prime number, then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$.

I have to prove that if $p\equiv 1,9 \pmod{20}$ is a prime number then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$. I consider the quadratic field $\mathbb{Q}(\sqrt{-5})$, with ring of ...
10
votes
5answers
599 views

Are numbers of the form $n^2+n+17$ always prime

Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it ...
10
votes
4answers
99 views

Show that $\mathcal{O}_K$ is not UFD with $K = \mathbb{Q}(\sqrt{-13})$

Let $K = \mathbb{Q}(\sqrt{-13})$. Show that its ring of integers $\mathcal{O}_K$ is not an UFD. $-13 \equiv 3 \bmod{4}$, so $\mathcal{O}_K = \mathbb{Z}\bigl[\sqrt{-13}\bigr]$. We will use the ...
10
votes
3answers
160 views

Something screwy going on in $\mathbb Z[\sqrt{51}]$

In $\mathbb Z[\sqrt{6}]$, I can readily find that $(-1)(2 - \sqrt{6})(2 + \sqrt{6}) = 2$ and $(3 - \sqrt{6})(3 + \sqrt{6}) = 3$. It looks strange but it checks out. But when I try the same thing for ...
10
votes
2answers
784 views

Factoring a number of complex integers?

Say you are given a number (ex: $377$) and you express it in a form that allows you to factor it over the complex integers: Notice, $377 = 16^2 + 11^2$ Thus: $(16 + 11i) $ and $(16 - 11i)$ Are ...
10
votes
2answers
578 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
10
votes
2answers
2k views

Relationship between Cyclotomic and Quadratic fields

Since $\varphi(p)=p-1$ is even the p'th cyclotomic field contains some quadratic field. Hecke says that in fact every quadratic field is contained by some cyclotomic field. What is this theorem ...
10
votes
1answer
513 views

Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
10
votes
3answers
2k views

What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?

I was recently reading about integral ring extensions. One of the first examples given is that $\mathbb{Z}$ is integrally closed in its quotient field $\mathbb{Q}$. Another is that ...
10
votes
2answers
490 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
10
votes
1answer
265 views

What does the German word “Zerlegungsautomorphismus” translate to?

I would like to know if any of our German friends can translate that word for me? Zerlegung is factorisation isn't it? So what is factorisation automorphism? This is taken from Deuring's paper “Die ...
10
votes
2answers
886 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
10
votes
3answers
302 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
10
votes
2answers
309 views

Classifying splittings of primes?

I was wondering what general strategies are available to figure out if a prime splits? I know for quadratic extensions there aren't too many possibilities for how a prime can split, so we essentially ...
10
votes
1answer
1k views

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
10
votes
2answers
831 views

Value of cyclotomic polynomial evaluated at 1

Let $\Phi_n(x)$ be the usual cyclotomic polynomial (minimal polynomial over the rationals for a primitive nth root of unity). There are many well-known properties, such as $x^n-1 = ...
10
votes
2answers
1k views

Finding the norm in the cyclotomic field $\mathbb{Q}(e^{2\pi i / 5})$

I'm doing one of the exercises of Stewart and Tall's book on Algebraic Number Theory. The problem concerns finding an expression for the norm in the cyclotomic field $K = \mathbb{Q}(e^{2\pi i / 5})$. ...
10
votes
1answer
221 views

Elementary proof that $\mathfrak{p}$ unramified in $L,L'$ implies unramified in $LL'$?

Let $K$ be a number field, let $\mathfrak{p}$ be a prime of $K$, and let $L,L'$ be extensions of $K$. Suppose $\mathfrak{p}$ doesn't ramify in either $L$ or $L'$. Is there a simple proof that ...
10
votes
1answer
281 views

Primes of ramification index 1 with inseparable residue field extension

I've been reading through Neukirch's Algebraic Number Theory, and I'm a little puzzled about a possibility with ramification of primes. As usual, let $\mathcal{O}_K$ be a Dedekind domain with field ...
10
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3answers
154 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
10
votes
3answers
213 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
10
votes
1answer
254 views

L-function for Dirichlet characters vs Hecke character

For a Dirichlet character $\chi: \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times} \to \mathbb{C}$, the Dirichlet L function is $$\prod_{q \neq p} (1 - \chi(q)q^{-s})^{-1}$$ If we lift this character to ...
10
votes
1answer
379 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
10
votes
4answers
185 views

Is the number of irreducibles in any number field infinite?

Are there infinitely many irreducibles in the ring of integers of any algebraic number field ? I tried to follow the same argument as we usually do for integers. Suppose there are finitely many ...
10
votes
1answer
250 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
10
votes
2answers
510 views

Cubic polynomials that generate the same extension?

For quadratic extensions we can easily determine when $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(\sqrt{b})$ by checking if $a/b$ is a square and this is easy to prove. I was wondering if there are any good ...
10
votes
1answer
238 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
10
votes
1answer
330 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
10
votes
1answer
94 views

Intersection of class number one fields

Let $F$ and $K$ be two number fields with class number one. How can one prove that the class number of $F \cap K$ is also equal to one. I have been trying to prove something like the intersection of ...
10
votes
0answers
80 views

What is the intuition behind Dirichlet's Class Number Formula? [closed]

As the title of the question suggests, what is the intuition behind Dirichlet's Class Number Formula being true? The Dirichlet Class Number Formula is$$h(\mathcal{O}_D) = -{1\over{D}} \sum_{n=1}^D ...
10
votes
0answers
111 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
10
votes
2answers
140 views

Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

Given a non-constant integral irreducible monic polynomial $f(x)$, the prime factors of its value at integers $x\in\mathbb{N}$ forms a set $\mathcal{P}(f)$. Is it possible that ...
10
votes
0answers
505 views

extension of Euler's totient function to number fields

It is well known that the Euler totient function $\varphi$ satisfies the formula $n = \sum_{d | n}\varphi(d)$. This follows for example from the fact that $\mathbb Z / n \mathbb Z$ can be written (as ...
10
votes
2answers
242 views

Diophantine equation $x^2 + 6(y+1)^2 = (y+2)^3$

I'm revising for exams and I've got stuck on an algebraic number theory question. The equation I'm trying to solve is $$ x^2 -2 = y^3, $$ and I was told to rewrite it as $$ x^2 + 6(y+1)^2 = (y+2)^3. ...
9
votes
3answers
1k views

Prime ideals in the ring of algebraic integers

Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$. It is known that $\mathcal{O}$ is a ...
9
votes
2answers
1k views

What does the discriminant of an algebraic number field mean intuitively?

If $E/F$ is a finite extension of fields and $\alpha_1,\ldots, \alpha_n$ is a basis of $E/F$, the discriminant of $\{\alpha_1,\ldots, \alpha_n\}$ is $$\det(\operatorname{Tr}_{E/F}(\alpha_i\alpha_j))$$ ...
9
votes
1answer
1k views

What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And ...
9
votes
2answers
264 views

Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$

On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with ...
9
votes
4answers
967 views

(Simple?) applications of Class Field Theory?

Does anyone know any simple/nice applications of class field theory? I would really like to find one related to diophantine equations, but anything you got would be good. Thanks
9
votes
3answers
75 views

Must an irreducible element in $\mathbb{Z}[\sqrt{D}]$ have a prime norm?

Let $D\in \mathbb{Z}$ where $D$ is not a perfect square. Prove that if $\alpha\in \mathbb{Z}[\sqrt{D}]$, and $\alpha= a+b\sqrt{D}$ with: $|a^2-Db^2| = p$, a rational prime, ...
9
votes
2answers
586 views

Every ideal of an algebraic number field can be principal in a suitable finite extension field

Let $K$ be an algebraic number field. Let $I$ be a non-zero ideal of the ring of integers $\mathcal{O}_K$ in $K$. By class field theory, there exists a finite extension(the Hilbert class field) $L$ of ...
9
votes
2answers
1k views

Ramification in a tower of extensions

I'm trying to make sense of all these theorems related to ramification. I was hoping someone would summarize these results. Assume we have: An extension $L/K$ and a some subextensions ...
9
votes
2answers
4k views

The units of $\mathbb Z[\sqrt{2}]$

How can I show that the units $u$ of $R=\mathbb Z[\sqrt{2}]$ with $u>1$ are $(1+ \sqrt{2})^{n}$ ? I have proved that the right ones are units because their module is one, and it is said to me ...
9
votes
2answers
328 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...
9
votes
4answers
481 views

Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every ...
9
votes
2answers
493 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 ...
9
votes
2answers
182 views

$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
9
votes
1answer
550 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 ...
9
votes
1answer
280 views

How does (21) factor into prime ideals in the ring $\mathbb{Z}[\sqrt{-5}]$?

The text of the exercise is the following: Show that $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain, and that the identities $21 = (4+\sqrt{−5})\cdot(4 − \sqrt{−5})$ and $21 = 3 · 7$ represent two ...