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Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

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12 views

Partial Injective Functions over Finite Fields

For a partial function $f: X \to F_q$, where $X \subseteq F_q$ be a subset of a finite field $F_q$, is there any criterions to judge whether $f$ is injective? For $X=F_q$, since any function from $...
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Does Weil's converse theorem holds for weight 1?

I'm reading Iwaneic's "Topics in classical automorphic forms". Now, I'm reading the proof the theorem that for any Hecke character $\xi$ of a quadratic field $K/\mathbb{Q}$, there exists a $\mathrm{GL}...
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6 views

The definition of polar density of a set of primes in a number field

In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes in a number field $K$, as follows: if for some positive integer $n$, the $n$-th power of the Dedekind zeta ...
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1answer
20 views

A question about Galois characters

Let $F$ be a number field and $\chi:\mathrm{Gal}(\overline{\mathbb{Q}}/F)\to\overline{\mathbb{Q}_\ell}^{\times}$ ($\ell$ a prime) a Galois character. My question is: Can we find a finite extension $K/...
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2answers
42 views

Find a maximal ideal in $R = \mathbb{Z}[\sqrt{−5}]$ containing the principal ideal (3)

I think I need to choose an element to pair it with 3, i.e. (3,x). But I don't see how to find such an element.
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Consider the order $\mathbb{Z}[\sqrt[4]{24}]$. Find all ideals of norm 100.

I have found that the ring of integers is $\mathbb{Z}[\alpha, \alpha^3/4]$ where $\alpha = \sqrt[3]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ ...
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1answer
41 views

$n\in \mathbf{N}$ such that a solution of $X^4+nX^2 +1$ is a root of unit

Consider $f_n(X)=X^4+nX^2 +1$ in $\mathbf{Q}[X]$. I found that for all natural $n$ such that $n\neq 2-m^2$ for a natural $m$, $f_n(X)$ is irreducible in $\mathbf{Q}$. Consider $K_n=\mathbf{Q}(x)= \...
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1answer
40 views

Proof of a stronger form of Chinese remainder theorem (12.3) in Neukirch

Let $\mathcal O$ be an order in a number field and ${\mathfrak a} \neq 0$ an ideal in $\mathcal O$. Then the theorem shows ${\mathcal O}/{\mathfrak a} = \oplus_{{\mathfrak p} \supseteq {\mathfrak a}} {...
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1answer
34 views

irreducibility of $(\zeta + \zeta ^{-1})$'s minimal polynomial over $\mathbb{Z}[x]$

This question relates to the next post, especially to Did's, answer (second answer in the post). Minimal Polynomial of $\zeta+\zeta^{-1}$ The answer gives a method to construct a monic polynomial $...
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1answer
12 views

Cocompact & discrete lattice

I don't understand a step in the proof of the following proposition: Let $\Lambda$ be a subgroup of a real vector space $V$ of finite dimension. Then $\Lambda$ is a full lattice if and only if $\...
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1answer
32 views

Log-concave coefficient polynomial which is not a chromatic polynomial

In this famous paper, June Huh proved Read's conjecture, which claims that coefficients of a characteristic polynomial $\chi_{G}(q) = a_{n}q^{n} - a_{n-1}q^{n-1} + \dots + (-1)^{n}a_{0}$ of any graph ...
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1answer
32 views

if $a$ is an algebraic integer and $m\in \mathbb{Z}$ then $a+m$ is an algebraic integer.

I tried to use newton binomial, if $p_a(x) = \sum_0^nb_kx^k$ , then $b_k(a+m)^k =b_ka^k +b_k\sum_0^{k-1}a^im^{k-i}$, denote $T_k=b_k\sum_0^{k-1}a^im^{k-i}$, one gets that $p_{a}(x)-(\sum_1^nT_k ) - ...
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1answer
48 views

the ring of integers: prove that (2) is a prime ideal and that it is a pid

Consider a real root $\alpha$ of $f(X)=X^3-3X+1$. Consider the ring of integers $A_K$ for $K=\mathbf{Q}[\alpha]$. I showed that the ideal $(1+\alpha)$ is prime in $A_K$ and that $A_K=\mathbf{Z}[\alpha]...
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1answer
30 views

Unramified extension of $L(\sqrt{\alpha})/L$

I am studying an article of Chaoli and I try to understand the following statement: If $L$ is a number field and $\alpha \in L^{\times}/(L^{\times})^{2}$ then, for an odd prime $p$, $L_{p}(\sqrt{\...
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1answer
51 views

A step in the proof of a version of Kummer's theorem

In a book I am reading (Cox, Primes), one of the exercises asks to prove the following version of Kummer's Theorem regarding factorization of primes in an extension: Suppose $L/K$ is Galois, $L=K(\...
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2answers
58 views

Class Number Calculation of a Real Quadratic Number Field

I am looking at the example below. Can anyone explain how they end up with the contradiction. Why do they reduce $a^2-65b^2$ modulo $5$ to show that it has no integer solutions?
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Why does the Frobenius-semisimplicity of a Weil representation not depend on the choice of the Frobenius element?

Definition: Let $K$ be a (non-Archimedean) local field and $k$ its residue field. A Frobenius element of the absolute Galois group $G_K$ is any element of $G_K$ which is a lift of the Frobenius ...
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If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if ...
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1answer
46 views

Showing that an ideal in $\mathbb{Z}[\sqrt{-21}]$ is principal

I have $\mathfrak{a}=(5,\sqrt{-21}-2).$ Can anyone tell me why $\mathfrak{a}^2$ is principal? I have multiplied the ideals out to obtain $$(5, 5\sqrt{-21}-10,-17-4\sqrt{-21}) $$ How does this reduce ...
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Lubin-Tate theory for non-maximal orders

Assume $O$ is an order in a $p$-adic field $K$, does there still exist a good theory of Lubin-Tate formal $O$-module? What field extension we will get by adding torsion points of such formal $O$-...
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1answer
54 views

Showing that an ideal is not principal in $\mathbb{Z}[\sqrt{-21}]$

I am trying to show that the ideal $$(2,\sqrt{-21}-1)(3, \sqrt{-21}) $$ is not principal in $\mathbb{Z}[\sqrt{-21}]$. Can anyone help with this?
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3answers
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Prove that $\sqrt{3\pm\sqrt{7}} \not\in \mathbb{Q}(\sqrt{3\mp\sqrt{7}})$.

I'm currently solving a fairly long exercise related to Galois theory in which I've come across having to prove that $\sqrt{3+\sqrt{7}} \not\in \mathbb{Q}(\sqrt{3-\sqrt{7}})$ and $\sqrt{3-\sqrt{7}} \...
3
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1answer
88 views

Nonarchimedean convergent power series

I would like to understand the second paragraph on the second page (marked page 320) of the article http://www.numdam.org/article/MSMF_1974__39-40__319_0.pdf on rigid analytic geometry by Michel ...
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1answer
59 views

Make a short exact sequence of abstract groups into a short exact sequence of topological groups (motivated by the Weil Group)

Let $1 \to H_1 \to G \to H_2 \to 1$ be a short exact sequence of abstract groups. Question: If $H_1$, $H_2$ have fixed topologies, can we endow $G$ with a topology such that the sequence above ...
4
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1answer
32 views

Field with vanishing Brauer group which is not $C_1$

In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$. Is there an example of a field which is not $...
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1answer
37 views

Chebotarev Density Theorem answers to factorization of polynomials

I've read of Chebotarev Density Theorem. The statement over $\mathbb{Q}$ is: Let $K$ be Galois over $\mathbb{Q}$ with Galois group $G$. Let $C$ be a conjugacy class of $G$. Let $S$ be the set of (...
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2answers
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Primes in $\mathbb{Z}[\sqrt{2}]$?

It's a well known result that the gaussian primes $\mathbb{Z}[i]$ can be characterized as normal primes $3 \bmod 4$ or normal primes $3\bmod 4$ times $i$. As well as $a+bi$ where $a,b$ are non-zero ...
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1answer
69 views

Characterization for the continuity of Weil representations

Let $K$ be a non-Archimedean local field and $W_K$ be the Weil group of $K$. We consider a representation $\rho: W_K \to \operatorname{GL}_n(\mathbb{C})$ between two topological groups. Here, $\...
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0answers
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Are all finite extensions of perfect fields cyclic?

I am not well trained in number theory. If there is any mistake, please straightly edit this question or devote me if the question is too trivial. According to https://en.wikipedia.org/wiki/...
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4answers
157 views

What other kinds of cubic integer rings are there?

Given an integer $n \in \Bbb{Z}$, we understand $\root 3 \of n$ to mean the number $x \in \Bbb{R}$ such that $x^3 - n = 0$. Then $\Bbb{Q}(\root 3 \of n) \subset \Bbb{R}$, right? The same then goes for ...
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Parametrization of cubic fields

A parametrization of quadratic fields is $\mathbb{Q}(\sqrt{m})$, where $m\ne1$ is a squarefree integer. That is, $\mathbb{Q}(\sqrt{m})$ is a quadratic field as $m$ varies, and all quadratic fields are ...
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0answers
65 views

What is the conductor of $ \mathbb Q(\sqrt 2 )/\mathbb Q?$ $4\mathbb Z$ or $8 \mathbb Z?$

By execise 6.8 of Childress's Class Field Theory, $\mathbb f(\mathbb Q(\sqrt 2)/ \mathbb Q)=8\mathbb Z$. But considering the norm of the local field about the place correspondent with 2, it should be $...
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Neukirch, Haar measure on Minkowski space

I would like to understand what Neukirch means when he writes down $$ vol_{canonical}(X) = 2^s vol_{Lebesgue}(f(X)) $$ (Neukirch, Algebraic Number Theory. Pg 31) I will write down the details at ...
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Sum of 2 squares implies efficient factorization

I'm concerning myself with factoring semi-primes and believe that if given a large semi-prime ($N$) one finds a non-trivial sum of squares representation: $$ x^2 + y^2 = N$$ Then one can ...
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1answer
35 views

Fourier Expansion of a function on $\mathbb A_k/k$

Let $k$ be a number field, and let $\mathbb A_k$ be the ring adeles of $k$. The quotient group $\mathbb A_k/k$ is compact, and the choice of a nontrivial character $\psi$ of $\mathbb A_k/k$ gives an ...
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2answers
60 views

Splitting of $2$ in a cubic extension

Let $L$ be the splitting field of $X^3-3X+1=0$. How does the prime $2$ split in $L$? I have figured out that either $2=\mathfrak{P}$ or $2=\mathfrak{P}\mathfrak{Q}\mathfrak{R}$. I guess it is the ...
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2answers
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If $p$ is prime, then $x^2 +5y^2 = p \iff p\equiv 1,9 $ mod $(20)$.

Let $p\neq 2,5$ be prime. I wish to show that: $x^2 +5y^2 = p \Leftrightarrow p\equiv 1,9 $ mod $(20)$. I proved to $\Rightarrow$ part, means $x^2 +5y^2=p \Rightarrow p\equiv 1,9 $ mod $(20)$. For $\...
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0answers
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Strong approximation and class number in the adelic setting

$\DeclareMathOperator{\GL}{GL}\DeclareMathOperator{\ord}{ord}\DeclareMathOperator{\SL}{SL}$I have a question about a proposition from Daniel Bump's book, Automorphic Forms and Representations. Here $...
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2answers
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7, 13 independent units in the local field $\mathbb{Q}_3$

Why do $7, 13$ generate a rank $2$ subgroup of the group of units of $\mathbb{Q}_3$? I.e. if $7^a= 13^b$ in the local field $\mathbb{Q}_3$ and $a, b$ are integers, then $(a, b)=(0,0)$. (This claim ...
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Prove that all algebraic numbers are included in any elementary substructure of $\mathbb R$

Let $A$ be an elementary substructure of $\mathbb R$ where $\mathbb R$ is $\langle \mathbb R,+,\cdot,0,1\rangle$ . Show that $A$ contains any algebraic number. What I tried to do was use the fact ...
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Primes lying above a prime in a Galois extension

Assuming $L$ is Galois over $K$, and that $O_L$ and $O_K$ are their respective rings of integers. Let $p$ be a prime ideal of $O_K$, is there a classification of the prime ideals laying above $p$ in $...
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3answers
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How to determine how many elements in $R / I$ where $R$ is ring of quadratic integers

This might be a very basic question for some of you. Indeed in $\textbf Z$, it's very easy. For example, $\textbf Z / \langle 2 \rangle$ consists of $\langle 2 \rangle$ and $\langle 2 \rangle + 1$. ...
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Differentiating a $p$-adic character

Let $L$ be a finite extension of $\mathbb Q_p$ with ring of integers $\mathcal{O}=\mathcal{O}_L$ and let $B_1(L):=\{z \in L \colon \vert z-1 \vert <1 \}$. Let $\widehat{\mathcal{O}}(L)_{\mathbb ...
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1answer
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Galois Groups in Ramification Theory

I had a slight confusion about Galois groups over a base field which is complete with respect to a discrete valuation. We know that there are irreducible polynomials such as $X^3+X^2+2X-8$ where ...
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1answer
59 views

Finding all primes above $x-a_i$ in the function field $y^2=(x-a_1)\cdots (x-a_n)$

This is a problem from Rosen's "Number Theory in Function Fields". Let $K=F(x,y)$ be a function field, such that $y^2=(x-a_1)\cdots (x-a_n)$, and all the elements $a_i$ are distinct. In $F(x)$ we have ...
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1answer
42 views

Is weight sum of Dirichlet character always an algebraic integer (up to power of $2$ and $3$)?

Let $\chi: (\mathbb Z/N\mathbb Z)^{\times} \rightarrow \mathbb C^{\times}$ be a character, consider $a=\frac{1}{N}\sum_{i=1}^N \chi(i)i$ where $\chi(n)=0$ if $n$ is not coprime to $N$. If $\chi$ is ...
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69 views

What is wrong with my conclusion (compositum of local fields)?

Let $K = \mathbb{Q}_2(\zeta_3)$ where $\zeta_3$ is a primitive third root of unity, and $F = \mathbb{Q}_2(\zeta_3,\sqrt[3]{2})$. Furthermore, let $L = \mathbb{Q}(\zeta_3,\beta)$ where $\beta$ is ...
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1answer
84 views

Computing degrees and ramification indices of some extensions of $\mathbb{Q}_2$

Let $K=\mathbb{Q}_2$ and $F = K(\zeta_3,\alpha)$ where $\zeta$ is a primitive third root of unity and $\alpha$ is a cubic root of $2$, i.e. $\alpha^3 = 2$.Let $K_1 = K(\zeta_3)$, $K_2 = K(\alpha)$ and ...
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57 views

Algebraic integers where all conjugates have absolute value at least 1

Let $\alpha$ be an algebraic integer with minimal polynomial $f$. Is there some natural condition on $f$ to guarantee that all Galois conjugates of $\alpha$ have absolute value at least $1$? ...
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2answers
305 views

Countability of Algebraic numbers proving the existence of some equality?

Let $\mathbb{A}$ denote the algebraic numbers. Consider the set of numbers defined as follows: $$S=\{2^{k_0}3^{k_1}5^{k_2}\ldots\;|\;k_0, k_1, k_2\ldots\in\mathbb{Q}\}$$ where $2,3,5\dots$ are the ...