Questions related to the algebraic structure of algebraic integers

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12
votes
1answer
281 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
45
votes
5answers
1k views
10
votes
1answer
278 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 ...
4
votes
2answers
83 views

Let $p$ be an odd prime. $p$ and $(1-\zeta_p)^{p-1}$ are associates in $\mathbb{Z}[\zeta_p]$.

Let $p$ be an odd prime and $\zeta_p$ a primitive $p$th root of unity, that is a $p$th root of unity other than 1. I need to show that $p$ and $(1-\zeta_p)^{p-1}$ are associates in ...
3
votes
1answer
165 views

A proof in Janusz Algebraic Number Field

I can't understand Lemma 6.2 from the Janusz book Algebraic number fields, pag. 26, that says: Let $A\subset B$ be integral domains with $A$ integrally closed and $B$ integral over $A$. If ...
1
vote
1answer
73 views

fraction field of the integral closure

Let $R$ be a domain, $K$ the field of fractions of $R$ and $L$ a finite field extension of $K$. Denote with $R'$ the integral closure of $R$ in $L$. Is it always true that $L$ is the field of ...
11
votes
3answers
646 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
2
votes
0answers
41 views

Additive characters of Quotient ring

Suppose $K$ is finite Galois extension of rationals, and let $\mathcal O$ be the ring of integers of $K$, and finally suppose $\mathfrak m$ is an integral ideal of $\mathcal O$, I want to classify ...
3
votes
1answer
54 views

Express in terms of familiar arithmetic functions

How can I express the sumation $$h_k(n)=\sum_{d|n, k|d}\mu (d)$$ in terms of familiar arithmetic functions, where $k\in \mathbb{N} $ is fixed?
1
vote
0answers
50 views

Is it always possible to find primes $p, q$ such that $\left(\dfrac{p}{q}\right)_n=\left(n,\dfrac{p-1}{f}\right)=1$?

I first provide a background, or the context, where this question arises. Skip it if one wants so. Background: In the book The Genus fields of algebraic number fields by Ishida, one finds the ...
6
votes
2answers
503 views

Showing that $\mathbb Q(\sqrt{17})$ has class number 1

Let $K=\mathbb Q(\sqrt{d})$ with $d=17$. The Minkowski-Bound is $\frac{1}{2}\sqrt{17}<\frac{1}{2}\frac{9}{2}=2.25<3$. The ideal $(2)$ splits, since $d\equiv 1$ mod $8$. So we get ...
7
votes
2answers
266 views

Solve: $x^2-py^2=q$

Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers. It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol. ...
5
votes
2answers
247 views

Are algebraic numbers analogous to group elements with finite order?

Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory? I thought this is the case since requiring an algebraic number $\alpha$ to be ...
2
votes
1answer
247 views

fractional ideals in the localization of a Dedekind

I'm reading Janusz, Algebraic number fields, 1973, pag.16-17, where defining a fractional ideal of a Dedekind domain $R$. A fractional ideal of $R$ is a non-zero finitely generated $R$-submodule ...
5
votes
2answers
182 views

Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$

I'm studying for my algebra quals this fall and keep encountering problems like the following: List all the ideals of $\mathbb{Z}[x]/(16, x^3)$. or List all the primes of ...
4
votes
2answers
194 views

Endomorphisms of the multiplicative formal group law

Is there a simple description of the ring of endomorphisms $\mathrm{End}(\mathbb{G}_m)$ of the formal group law $$\mathbb{G}_m(X,Y) = X + Y + XY,$$ at least over a ring of characteristic zero? I'm ...
2
votes
3answers
593 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
5
votes
2answers
664 views

Is the inverse of a fractional ideal still fractional?

Let $R$ be a Dedekind domain, $K$ the field of fractions of $R$, $\mathfrak{m}$ be a fractional ideal of $R$, i.e. a non-zero finitely generated $R$-submodule of $K$. We define ...
6
votes
3answers
1k views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
8
votes
3answers
304 views

Solve $x^3+x \equiv 1 \pmod p$

Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions. We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$ We can prove that $(1)$ cannot ...
5
votes
1answer
497 views

How to determine a Hilbert class field?

I tried to solve the exercise VIII.XX in Number Fields by Marcus. It asks to find the Hilbert class field of $Q(\sqrt m)$ for $m=-6,-10,-21,-30$. And the emphasis of this question is on the first two. ...
1
vote
2answers
127 views

Nonreal units in totally imaginary number fields

Suppose we are given a totally imaginary number field $L$ of degree greater $2$, is it possible that all units of $L$ lie in $\mathbb R$? This is true for almost all quadratic imaginary number fields, ...
9
votes
1answer
2k views

Vague definitions of ramified, split and inert in a quadratic field

Our lecturer defined the following: Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then (1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$ ...
2
votes
1answer
85 views

Euclidean algorithm in lattices in $\mathbb{C}$

Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. I want to prove that, for every $\alpha,\beta\in R,\beta\neq 0$, there exist $\gamma,\delta\in R$ such that $\alpha=\gamma\beta+\delta$, with ...
2
votes
1answer
105 views

How to find $\sup(\{|x-y|_p : x,y\in B(0;r)\})$

Just to clarify the notation and the question: Working in p-adic space $\mathbb{Q}_p$, we have the norm $|x|_p=p^{-ord_p(x)}$ and we define the metric over this space as $d(x,y)=|x-y|_p$. We are ...
2
votes
1answer
70 views

$\mathbb{Q}[\sqrt a_1,\ldots,\sqrt a_k]$ vs. $\mathbb{Q}(\sqrt a_1,\ldots,\sqrt a_k)$

Call an algebraic number polyquadratic if it can be expressed as the sum or difference of a finite number of square roots of rational numbers. (This definition follows Conway-Radin-Sadun rather than ...
1
vote
0answers
45 views

How to measure the failure of Hasse norm theorem?

We know that the failure the unique factorisation is measured by the ideal class-group, that of the local-global principle depends upon the Tate-Shafarevich group. Then I thought: what should be ...
4
votes
1answer
133 views

Finiteness of ideal of given norm

I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$. I know there are "standard proofs" (eg How many elements ...
3
votes
2answers
80 views

Is it true that $\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap \mathbb{Q}$?

I know $\mathbb{Z}_{(p)}\subset \mathbb{Z}_{p}\cap \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at prime ideal $(p)$ and $\mathbb{Z}_p$ is the set of p-adic integers. I ...
4
votes
1answer
68 views

On Selmer's curve

I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
9
votes
1answer
503 views

Intuition for Krasner's Lemma

From Milne's Algebraic Number Theory, we have (he assumes that $K$ is complete with respect to a discrete nonarchimedian absolute value, but I don't know where the discrete part is being used) Let ...
1
vote
0answers
44 views

Artin L- Function properties

I'm trying to understand the proof of one of the properties of the Artin L-function. I have the following doubts; Why take on $f_i =|G_{P_i}: H_{P_i}I_{G,P_i}|$, $H_{P_i}I_{G,P_i}$? and not only ...
5
votes
3answers
438 views

Proof of Hasse-Minkowski over Number Field

Can anyone point me towards a good reference which contains the proof of the Hasse-Minkowski theorem of quadratic forms over a number field? Serre's "A Course in Arithmetic" has a self contained proof ...
1
vote
0answers
107 views

Solve the equation $x^4+y^4=d*z^2$

Solve the equation:$$x^4+y^4=d*z^2,$$ where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer. I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is ...
7
votes
1answer
204 views

p-adic modular form example

In Serre's paper on $p$-adic modular forms, he gives the example (in the case of $p = 2,3,5$) of $\frac{1}{Q}$ and $\frac{1}{j}$ as $p$-adic modular forms, where $Q = E_4 = 1 + 540\sum ...
6
votes
1answer
675 views

Norm of ideals in quadratic number fields

I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
3
votes
1answer
217 views

Computing the class number of $\mathbb{Q}(\sqrt{1533157})$

I am trying to compute the class number of $\mathbb{Q}(\sqrt{1533157})$ in Magma. Can anyone explain why it's taking so long to compute? I'm currently running Magma V2.18-7. Below is my code: ...
3
votes
1answer
525 views

Definition of nebentypus in $L$-functions.

In Iwaniec and Kowalski, the term nebentypus is mentioned several times in the book. Every time it seems to just refer to a character $\chi$. Since I don't see the authors defining nebentypus, can ...
0
votes
1answer
238 views

Cyclotomic euclidean number fields

I´m here because I want to repeat my question about the norm-euclidean algorithm in a particluar cyclotomic integer ring. Let $L=\mathbb{Q}(\zeta_{32})$ and $A=\mathbb{Z}[\zeta_{32}]$. On the page ...
2
votes
1answer
90 views

Whether a domain is Dedekind or not

We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
4
votes
3answers
148 views

Show that there exists $f ∈ \mathbb{Z}$ such that $f^2 + f +1 ≡ 0 \pmod p$.

Let $p ≡ 1 \pmod 3$ be a prime. Show that there exists $f \in \mathbb{Z}$ such that $f^2 + f +1 \equiv 0 \pmod p$. I know the first few primes of this form are: $7,13,19$ So for example $p=7$ we ...
7
votes
5answers
515 views

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
4
votes
1answer
195 views

Fermat's Last Theorem in multiple variables

I was wondering if there was anything we could say about when, given $m$, $\exists n (\forall x_1,\dots,x_m \in \mathbb{N} ( x_1^n + x_2^n + \dots + x_{m-1}^n \neq x_m^n))$ Fermat's Last Theorem ...
4
votes
2answers
127 views

Proving a factorization of ideals in a Dedekind Domain

Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
4
votes
1answer
130 views

Why does this class group seem inconsistent?

I'm computing a class group for an imaginary quadratic field and something seems wrong. Let $\delta=\sqrt{-29}$ and let $R$ be the ring of integers in $\mathbb Q[\delta]$. From ...
6
votes
2answers
280 views

Ideals in a Dedekind domain localized at a prime ideal

Let $R$ be a Dedekind domain, let $\mathfrak{i}$ be a non-zero ideal of $R$. By factorization theorem we can write $$\mathfrak{i}=\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$$ for distinct non-zero ...
1
vote
2answers
267 views

How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory

I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units ...
2
votes
1answer
147 views

Stable points and the fundamental domain of the modular group

Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain. How ...
2
votes
1answer
313 views

If $x^p +y^p = z^p$ and $xyz \neq 0$, then $p$ divides $x$ or $y$ or $z$?

I am working on an exercise: If $x^5 +y^5 = z^5$ and $xyz \neq 0$, then $5$ divides at least one of $x$, $y$ or $z$. I am thinking that the answer involves an application of Kummer's theorem, but I'm ...
7
votes
1answer
125 views

Implications between $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ and $(\frak{a})$ + $(\frak{b})$= $(1)$

In a general commutative ring, $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ does not imply ($\frak{a}$) + ($\frak{b}$) = ($1$); whereas ($\frak{a}$) + ($\frak{b}$) = ($1$) does imply $(\frak{a} ...