Questions related to the algebraic structure of algebraic integers

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4
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1answer
35 views

What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
7
votes
1answer
54 views

What is the smallest $d$ such that $4$ has more than one distinct factorization in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?

Or if there is no such $d$, how do I prove it? Obviously there is no point to looking for this in an UFD. I've looked in other rings, and each time I think I found it, I divide one of the factors by ...
0
votes
0answers
16 views

Degree of the separable closure of the residue field of a complete field

I'm trying to prove a corollary on ramified extensions but I don't know if my thoughts are true. Let $L$ be a finite extension of a complete discrete valuation field $F$, let ...
1
vote
0answers
36 views

$p$-divisible group of tori

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the ...
6
votes
3answers
110 views

Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 ...
2
votes
1answer
24 views

Problem with the hyperelliptic equation

Suppose $K$ is an algebraic number field with $ [ K : \mathbb{Q} ] = d $. $X, Y , \alpha_1 , \ldots \alpha_n $ are in $O_K$ , i.e. are integral over $\mathbb{Z} $. Suppose that we have the following ...
2
votes
3answers
31 views

Given all the multiples of a prime number $p \in \mathbb{Z}$, is $p\mathbb{Z}$ an ideal of $\mathbb{Z}$?

So I'm having a little trouble understanding the concept of an ideal. The book gives the "classic example" of $2\mathbb{Z}$, the even integers, saying these form an ideal. Would I be correct in ...
2
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0answers
55 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
0
votes
0answers
52 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
3
votes
0answers
41 views

Algebraic integers of the form $2\cos (2\pi r)$ and Kronecker's Theorem

Problem: Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are all real and lie in the interval $[-2,2]$. Prove that each root of $P$ has the form $2\cos (2\pi r)$ for some $r\in ...
2
votes
0answers
146 views

$e=1$ in Theorem 30 from Marcus book “number fields”

Theorem 30 in Marcus book states that, if $p\in\mathbb Z$ is an odd prime and $q$ is a prime $\neq p$, then, fixing $d$ as a divisor of $p-1$ we have that $q$ is a $d$-th power $\operatorname{mod}q$ ...
1
vote
0answers
45 views

Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
0
votes
0answers
35 views

Zeros of a complex function

Consider the function $$f(x)= \sum_{j=1}^n b_j e^{i a_jx},$$ where $a_j,b_j$ are algebraic numbers. Denote $A=\{f(x)| x\in \mathbb{R}_{\geq 0}\}$, i.e., $A=f([0,\infty))$. Does this hold? $ 0\in ...
0
votes
1answer
74 views

Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...
0
votes
1answer
44 views

Summation and product over $k$ with $k$ prime to $n$ sought

I just come to a standstill with the following two formulas. If $$E_n=\lbrace k\mid 1\le k\le n\ \&\ (k,n)=1\rbrace$$ then I hope for a closed formula $f(n)$ for those $$\sum_{E_n}k$$ ...
2
votes
0answers
21 views

Two definitions of Hecke characters

Let $K$ be a number field and $\mathfrak{m}$ an integral ideal of $\mathcal{O}_K$. Denote by $I(\mathfrak{m})$ the group of fractional ideals prime to $\mathfrak{m}$, and by $P(\mathfrak{m})$ the ...
0
votes
1answer
66 views

The quotient of the ring of integers by an ideal always seems to have the same structure

Can the following be proven or does there exist a counter example: The quotient of the ring of integers of a numnber field by an ideal is the direct sum of rings that are either isomorphic to ...
1
vote
1answer
17 views

Exceptional Set and Schanuel's conjecture

I was reading an article about transcendental funtions (Algebraic values of transcendental functions at algebraic points, by Huang, J., Marques, D., Mereb, M.). The authors gave an example that says: ...
2
votes
0answers
22 views

Prove that $\mathfrak{p}$ is totally split in $L/K$ and $L'/K$ $\Rightarrow$ totaly split in $LL'/K$

Assume that $K$ be a number field and $L/K$, $L'/K$ are two separable extensions. Now let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$. Then if $\mathfrak{p}$ is totally split ind $L$ and $L'$, ...
1
vote
1answer
31 views

Showing the center of an endomorphism ring is a direct summand

I am reading A. Fröhlich's Formal Groups, and I am working on the proof that if $F$ is a formal group defined over a separably closed field $k$ of characteristic $p$, then the endomorphism ring $E$ of ...
0
votes
1answer
46 views

The minimum number of digits after the floating-point, which uniquely identify every irrational square root

Let the following: $B:$ a natural number larger than $1$ $S:$ a set of irrational numbers in the range $(0,1)$ represented in base $B$ $L:$ the minimal prefix length which uniquely identifies every ...
0
votes
0answers
25 views

Number of prime ideals that contain a non zero ideal

In the proof of proposition (12.3) of Neukirchs Algebraic Number Theory, we use the fact that for a one-dimensional noetherian integral domain, there are only finitely many prime/maximal ideals that ...
1
vote
1answer
22 views

Bound on integral solutions to $ar^2+bs^2=m$

The problem is as follows. Let $m$ be a fixed integer. Let $a,b\geq0$ be integers such that $(a,b)=1$ and both $a$ and $b$ are square-free. I want to show that the set $\{r,s\in\mathbb ...
0
votes
0answers
24 views

Why is a mapping $H \sigma G_{\mathfrak{P}} \mapsto \sigma\mathfrak{P} \cap L$well-defined bijection?

Assume that $K$ is a number field with a ring of integers $\mathcal{O}_K$. Let $L/K$ be an arbitary separable extension, and embed it into a Galois extension $N/K$ with Galois group $G = ...
1
vote
0answers
100 views

How to prove that an ideal in the class group is not principal

Let $\theta$ be a root of the polynomial $p=x^3+6x-1$, and $K=\mathbb{Q}\left[\theta\right]$ the field generated by it, and $\mathcal{O}_K$ its ring of integers with integral basis ...
3
votes
1answer
37 views

Literature on fields $\mathbb{Q}(\sqrt[4]{D})$

Can anyone direct me to literature discussing extension fields of the form $K = \mathbb{Q}(\sqrt[4]{D})$ where $D$ is squarefree? I'm particularly interested in results regarding the class and unit ...
7
votes
1answer
57 views

An extension of an algebraic number field which makes an integral ideal $I$, a principal ideal

I want to show that, given an ideal $I \subseteq \mathcal O_K$ (where $K/\mathbb Q$ is an algebraic number field), there is a finite extension $K'/K$ such that, $I\mathcal O_{K'}$ becomes a principal ...
4
votes
3answers
89 views

Does there exist a cubic polynomial $f(x)$ such that $f(x)\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$?

As we know that $f(x)=x^2+1\equiv0 \pmod p $ has no integer solutions if $p\equiv 3\pmod 4$, does there exist a cubic polynomial $f(x)=ax^3+bx^2+cx+d~(a,b,c,d \in\mathbb Z,a\neq 0) $ such that ...
0
votes
1answer
43 views

Dedekind rings which are UFDs but not PIDs?

I just have a really quick question of an example that I was trying to come up with. Are there any number rings which are UFDs but not PIDs?
0
votes
0answers
31 views

$A\alpha_1+\dots +A\alpha_n$ is a $B$-module

Let $A$ be an integrally closed domain with quotient field $K$ and $L/K$ a finite separable field extension. We denote with $B$ the integral closure of $A$ in $L$. Now let $\alpha_1,\dots,\alpha_n\in ...
1
vote
1answer
45 views

Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
2
votes
1answer
51 views

Algebraic Integers in $\mathbb{Q}(\sqrt{m})$ and Norms on them

I'm having a problem with a section of Niven's book the Theory Of Numbers. I am trying to show: If an integer $\alpha \in \mathbb{Q}(\sqrt{m})$ is neither zero nor a unit, prove that ...
3
votes
1answer
43 views

Non unique factorization domains with prime factorizations with differing number of primes

As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations ...
3
votes
0answers
48 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
28
votes
5answers
794 views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
3
votes
2answers
54 views

Does it hold that the $p$-adic completion of the integers equals the completion of the localization in $p$?

maybe this is a stupid question, but I could not solve it even for the ordinary integers $\mathbb{Z}$. Furthermore, I don't have to much knowledge on algebraic number theory and ramifications. Let ...
3
votes
3answers
44 views

If algebraic $a$ has degree $n$, so does $-a$

I feel like the best way to move forward is to use a contradiction proof. Since $a$ is algebraic, and is of degree $n$, it has a minimal polynomial of degree $n$, so we can write ...
6
votes
2answers
66 views

minimal polynomial given an algebraic number

I am trying to find the minimal polynomial for the algebraic number $1+\sqrt{2}+\sqrt{3}$. My original thought was just let $\alpha=1+\sqrt{2}+\sqrt{3}$. The method I use though seems very ...
5
votes
1answer
67 views

Class group of $\mathbb{Q}(\sqrt[4]{-2})$

I would like to show directly that $C(K)$ is trivial, where $K = \mathbb{Q}(\sqrt[4]{-2})$. Write $\delta = \sqrt[4]{-2}$. It is pretty easy to see that $\mathcal{O}_K = \mathbb{Z}[\delta] = R$. Then ...
3
votes
2answers
63 views

explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by ...
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vote
2answers
53 views

Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
2
votes
2answers
41 views

Finding ideal representatives in the class group of $\mathbb{Q}(\zeta_{23})$

I know that $\mathbb{Q}(\zeta_{23})$ has class number 3, and I am wondering how I can find ideal representatives of the two nonprincipal classes in the class group. I have tried looking at examples ...
1
vote
1answer
52 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
6
votes
2answers
121 views

Finding non-negative integers $m$ such that $(1 + \sqrt{-2})^m$ has real part $\pm 1$.

I believe that the integers $m$ with $(1+\sqrt{-2})^m$ having real part $\pm 1$ are $0, 1, 2$ and $5$, but I'm having trouble proving it. Write $$a_m = \Re((1+\sqrt{-2})^m) = \frac{(1 + \sqrt{-2})^m ...
46
votes
4answers
3k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
3
votes
1answer
52 views

Question on number theory ( related to (Z/p^rZ)* group )

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
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vote
0answers
63 views

Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
1
vote
0answers
49 views

Trivial Rost-Motive of a quadric

Let $q$ be the anisotropic,quadratic form of rank two corresponding to $\alpha = d(q) \in H^1(k,\mu_2)$. In his lecture notes "Topics in quadratic Forms" Vishik writes: For $n=1$ we get the ...
1
vote
1answer
22 views

Extensions of nonarchimedean valuations

This is a question from Janusz 'Algebraic Number Theory'. Let $R$ be a DVR with maximal ideal $\mathfrak p=\pi R$. Let $K$ be the quotient field of $R$ and $\mid\cdot\mid_{\mathfrak p}$ the ...
2
votes
1answer
54 views

Strong approximation theorem for Dedekind Domains

This is a theorem in "Maximal Orders" by Reiner. Page 48 stated without proof. And is said to be an easy consequence of The Chinese remainder Theorem. I am attempting to prove the theorem and need a ...