Questions related to the algebraic structure of algebraic integers

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0
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1answer
16 views

How to show that $\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$

How to show that $\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$, where $p$ is an odd prime, $\eta$ is $p$-th roots of unity and $x,y$ are integers. It could be reduced to the form ...
3
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2answers
34 views

Maximal unramified extension of a global function field

Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?
2
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1answer
43 views

Two disjoint number fields $K$, $L$ such that $(\mathrm{disc}({\cal O}_K), \mathrm{disc}({\cal O}_L))\neq 1$ but ${\cal O}_L{\cal O}_K={\cal O}_{KL}$

I know that if two disjoint number field $K$, $L$ are such that $(\operatorname{disc}(\mathcal{O}_K), \operatorname{disc}(\mathcal{O}_L))= 1$ then $\mathcal{O}_L\mathcal{O}_K=\mathcal{O}_{KL}$. It is ...
2
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0answers
24 views

Prove that $\mathbb{A} \cap \mathbb{Q}(\sqrt{2},\sqrt{-3})$ is a PID.

While self-studying algebraic number theory, I came across the following problem: Prove that $\mathbb{A} \cap \mathbb{Q}(\sqrt{2},\sqrt{-3})$ is a PID. where $\mathbb{A} \cap ...
5
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3answers
80 views

How to prove there are no solutions to $a^2 - 223 b^2 = -3$.

As the title suggests, I'm trying to prove that there are no solutions to $a^2 - 223b^2 = -3$ (with $a,b\in \mathbb{Z}$). Ordinarily, taking both sides $\mod n$ for some clever choice of $n$ proves ...
2
votes
1answer
15 views

profinite completion of a ring of integers in a global field

It is well known that the profinite completion of the integers, $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ is, by the Chinese remainder theorem, isomorphic to $\prod_p \mathbb{Z}_p$. I ...
3
votes
1answer
24 views

Non-archimedean exponential valuation and integral closure

I am trying to solve the following problem from Neukirch's book on ANT: Let $L|K$ be a finite field extension, $v$ a nonarchimedean exponential valuation, and $w$ an extension to $L.$ If ...
5
votes
1answer
96 views

When $\cos(\theta) = 1/8$ it's easy to show $\theta$ is an irrational angle. Is it algebraic?

Along the lines of my lines of my previous question about irrational angles "$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?", I was working on ...
2
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1answer
27 views

Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
2
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1answer
44 views

Associate a discrete valuation ring to a field $k$.

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero? I ...
1
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1answer
21 views

Galois group of maximally tamily ramified extension over the maximally unramified extension of a global function field F

Suppose $F$ were a local nonarchimedean field of characteristic zero of residue characteristic $p$. Let $F^{un}$ the maximally unramified extension. Let $F^{un}\subset F^{tame}$ be the maximally tame ...
1
vote
1answer
45 views

Is every algebraic integer a sum of roots of $x^n - a$?

A complex number is said to be an algebraic integer if it is a root of a monic polynomial with integer coefficents. For example any root of the polynomial $x^n - a$ for $a \in \mathbb{Z}$ is an ...
0
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0answers
44 views

Are these functions characters?

In Ireland and Rosen are mentioned the functions $f,g:\mathbb{F}_{p^f}^*=G\to \mathbb{C}$, with $$f:x \mapsto \zeta_p^{x+x^2+\cdots + x^{p^{f-1}}}$$ $$g:x \mapsto \left( \frac{x}{P}\right)_m,$$ ...
4
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3answers
219 views

Find the number of integral solutions of $(x,y)$

Given this equation: $4x^3+5=y^2$ Find the ordered pairs of $(x,y)$ where $x,y\in Z$
1
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1answer
39 views

Good introductory book for Probabilistic Number Theory

I have a decent high school knowledge of Elementary Number Theory and it is also a subject I love to study. I have a good background in Real Analysis (not Complex Analysis) and Abstract Algbera. I ...
1
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0answers
27 views

Properties of a Semi-modulo! operation

Let $A$ be an integer with its representation in base $p$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p}$$ We know $A\equiv (a_n\ldots a_1a_0)_{p}\pmod ...
4
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1answer
96 views

Salem numbers and Lehmer's decic

Given, $$x^{12}-x^7-x^6-x^5+1 = 0\tag1$$ This has Lehmer’s decic polynomial as a factor, $$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1=0\tag2$$ hence one of its roots is the smallest known ...
1
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1answer
30 views

Quadratic reciprocity in the case $a=-1$

I am reading the proof the for odd prime $p$, $$ \left ( \frac{-1}{p} \right)_2 = (-1)^{\frac{p-1}{2}} = \begin{cases} 1 \hspace{2mm} \text{for} \hspace{2mm} p \equiv 1 \operatorname{mod} 4 \\ -1 ...
0
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0answers
26 views

On Period of Linear Recurring Sequences modulo $P^e$

If a sequence $ X_0,X_1,X_2,\ldots$ is defined in terms of an initial set $ X_0,X_1,X_2,\ldots ,X_{k-1} $ by the recurrence relation $$ X_{n+k}= ...
4
votes
0answers
123 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
3
votes
1answer
119 views

Is the real number $\sqrt{6}$ in $\mathbb{R}$ equal to the 5-adic number $\sqrt{6}$ in $\mathbb{Q}_5$?

My question is as in the title. That is, consider solving the equation $x^2-6=0$ in $\mathbb{R}$ and in the 5-adic field $\mathbb{Q}_5$ respectively. We obtain one $\sqrt{6}\in\mathbb{R}$ and one ...
2
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0answers
36 views

Basic numbers and minimisation

Given $a$ and $b$ find $c$ and $d$ such that $bc-ad$ is least and greater than zero? Also $a,b,c,d$ are integers and all lie inside a given range i.e. $[0, n]$. For example, if $n=50, a=48$ and ...
4
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1answer
43 views

$\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$, adeles.

Let $p$ be a prime number. How do I show that $\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
2
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0answers
40 views

Norm map and extension of idele class groups

It's a classic fact in class field theory that for an extension of number fields $L/K$, we have the norm map on idele class groups $\mathcal{N}:C_L\to C_K$ descended from the map on ideles given by ...
0
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1answer
11 views

Reference request: principalization theorem

Let $K$ be a number field, and $\mathbb{I}_K$ the group of ideles. The Hilbert class field $M$ of $K$ is the class field of the open subgroup $H = K^{\ast} \mathbb{I}_K^{S_{\infty}}$, where ...
4
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0answers
38 views

Motivation behind the Archimedean norm on number fields .

It is easy to justify the use of non-archimedean norms on number fields from an "inside view" as arising from the prime ideals and are therefore clearly useful a priori. However, it seems to me that ...
1
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1answer
57 views

Proof of direct sum of ideal class group of Neukirch book

In books Neukirch, Algebraic Number Theory. I don't understand. 1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak ...
3
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2answers
25 views

Classifying algebraic integers satisfying a positivity condition

Let $a $ be an algebraic integer such that $1/a$ is also an algebraic integer belonging to the ring of integers of $\mathbb {Q}(a) $. Then, what is the condition for $a $ to satisfy: For any ...
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0answers
19 views

Factorisation And Ideals

I have only a basic grasp of algebraic number theory. I understand the proofs of UF for the rational integers and the analogous proofs (using norms) of UF for Gaussian and Eisenstein integers, for ...
0
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1answer
378 views

Prime divisors of the conductor of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of ...
6
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1answer
63 views

Given $d \equiv 1 \pmod 4$, can $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ have a fundamental unit that is not a “half-integer”?

Also $d > 0$, $\mu(d) \neq 0$. I acknowledge the term "half-integer" is, at best, problematic. What I mean by it is that an integer in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ is of the form $\frac{a ...
6
votes
2answers
80 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
4
votes
1answer
31 views

Unramified Hecke character

I'm looking for a reality check here: Let $\chi$ be a character of $(F^\times\backslash \mathbb A_F^\times)^1$ where $F$ is a number field. Call $\chi$ unramified at a place $v$ if ...
2
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1answer
330 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
2
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0answers
25 views

Given $r \in \mathbb{N}$, prove there exist odd primes $p,q$ s.t. $p$ splits into $r$ primes in the $q$th cyclotomic field.

Given $r \in \mathbb{N}$, prove there exist odd primes $p,q$ s.t. $p$ splits into $r$ primes in the $q$th cyclotomic field. Let $\omega = e^{2\pi i /q}$. Then I know that $(p,q) = 1 \implies p ...
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0answers
562 views

Serge Lang Never Explains Anything Round II

I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote: Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian ...
2
votes
1answer
34 views

Factoring ideals in algebraic number rings using Dedekind's theorem

Let $K \subset L=K(\alpha)$ be a number field extension with rings of integers $\mathcal{O}_K$ and $\mathcal{O}_L=\mathcal{O}_K[\alpha]$ respectively. Let $\pi$ be a prime ideal in $O_K$, and let $F = ...
0
votes
2answers
58 views

Definition of algebraic integer

Can the definition of algebraic Integer The roots of polynomials, such as $x^3 + bx^2 + cx + d = 0$, with integer (or rational) coefficients. be accurately paraphrased as Any real or ...
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0answers
72 views

Neukirch's motivation for $p$-adic numbers

I've started reading Neukirch's Algebraic Number Theory book and at the beginning of Chapter II he starts his motivation for the $p$-adic numbers as follows: "The idea originated from the observation ...
6
votes
2answers
77 views

Existence of $\sqrt{-1}$ in $5$-adics, show resulting sum is convergent.

I know that to prove the existence of a square root of $-1$ in $\mathbb{Z}_5$, I can just plug $x = -5$ and $a = 1/2$ into the Taylor expansion$$(1 + x)^a = \sum_{n=0}^\infty \binom{a}{n} ...
0
votes
1answer
14 views

How to remove $1$ at position $x$ ( in base $B$) from a number represented in Base $10$

I was going through a solution on code chef in which we needed to remove a $1$ from a position say $x$ (in Base $B$) from a number in Base $10$ if the representation of that number in base $B$ had a ...
3
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2answers
2k views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...
0
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0answers
41 views

Representations of algebraic group ($S_{\mathfrak{m}}$)

I'm studying Serre's book "Abelian $\ell$-adic Representations and Elliptic Curves" and in chapter II $\S$2.4 we have this proposition: Consider $v$ a finite place of $K$ and $F_v \in Gal(K^{ab}/K)$ ...
1
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0answers
31 views

Product of complex numbers $m+in$ with $0 < m,n \leq N$

I am trying to look for a generalization of Stirling's formula to complex numbers. In the integer case: $$ \log n! = \sum_{k = 1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n$$ For the ...
2
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0answers
35 views

Online lecture Videos on Algebraic or Analytic Number Theory or Sieve Theory [closed]

I found some lectures on youtube but I need something which starts from the basics.Any help will be truly appreciated.
2
votes
1answer
30 views

idelic ray class group modulo $\mathfrak{m}$

I'm studying the idele group $\mathcal{I}$ for a number field $K$. My definition of the ray class group attached to a modulus $\mathfrak{m}$ is $$\mathcal{C}_{\mathfrak{m}}= ...
9
votes
0answers
76 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 ...
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0answers
29 views

Determinant of the change of basis for fractional ideal

Let $A$ be a fractional ideal of some number field extension $K:\Bbb Q$. Let $\omega_1, \dots ,\omega_n$ be a $\Bbb Z$ basis for $\mathcal O_K$ and let $\alpha_1, \dots ,\alpha_n$ be a $\Bbb Z$ basis ...
2
votes
1answer
50 views

The criteria for two abelian extensions to be embedded

Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help. Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if ...
8
votes
1answer
73 views

Zeta function, $\mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$

Let $A = \mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$. My question is, what is the easiest way to see that$$\zeta_A(s) = {{1 + 2 \cdot 5^{-s} + 5^{1 - 2s}}\over{1 - 5^{1 - s}}}?$$Much thanks in advance. ...