Questions related to the algebraic structure of algebraic integers

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265 views

Prove generalisation of the Tower Law

I need to prove the following generalisation of the Tower Law: Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if $V$...
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49 views

Solve $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$

Prove that for every natural number $m$ there is a natural solution for the eqation $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$ Beside the typical inequality I can't get nothing ...
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94 views

A nonprincipal ideal and a nonprime irreducible in $\mathbb{Z}[\sqrt{-17}]$

The problem asks to find a nonprincipal ideal and a nonprime irreducible in $R = \mathbb{Z}[\sqrt{-17}]$. Since $-17 \equiv 3 \pmod 4$, $R$ is the ring of integers of $\mathbb{Q}(\sqrt{-17})$. I ...
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66 views

Existence of an ideal with norm 13 in $\mathcal{O}_K$

If $K=\mathbb Q(\sqrt{15})$ then $\mathcal O_K=\mathbb Z[\sqrt{15}]$. $13$ is prime in $\mathcal O_K$. Question: Does there exist and ideal $\mathfrak {a}$ of $\mathcal{O}_K$ such that $N (\mathfrak ...
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53 views

Compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$

How do I compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$? For example, $\alpha = \sqrt[3]{-19}$ and $\beta = (\alpha^2 - \alpha + 1)/3$ satisfy $(\alpha + 1)\beta = -6$,...
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132 views

Galois theory on curves

Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line $\mathbb{P}^{1}(\...
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87 views

Algebraic number field with non trivial integral basis

So far I have only seen extensions of $\mathbb{Q}$ with "trivial" integral basis. Meaning that the integral basis is the most natural one e.g. the integral basis for $\mathbb{Q}(\sqrt[3]{2})$ is just $...
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48 views

Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
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108 views

Splitting of Primes in a Given Field

Find how $p=2,3,5,7$ splits in $\mathbb{Q}(\sqrt{-5})$ (i.e. find those $e_i,f_i$ for $1 \leq i \leq r$). Can somebody please explain how this is done? My attempt is the following: Let K = $\mathbb{...
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60 views

Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields. Suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
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52 views

Prove that $\mathbb{Z}[\zeta_{p} + \zeta_{p}^{-1}]$ is the ring of integers of $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1})$

I'm a bit at a loss about what I can say in this situation. Do I have to show that $\zeta_{p} + \zeta_{p}^{-1}$ form an integral basis ? If I do, I have no idea how to do it. If not, can I use the ...
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34 views

Terminology — lying over something

I just came across reading something like this: 'Let $\phi\in \text{Gal}(L/K)$ lie above $Frob\in \text{Gal}(K^{un}/K)$.' Where $Frob$ is the Frobenius automorphism and $K^{un}$ is the maximal ...
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If a ring has its field of fraction as algebraic number field $K$, would this ring be $O_K$?

Suppose that ring has its field of fraction as algebraic number field $K$. Would this ring then be $O_K$, ring of integers? Also, for $O_K$, would subring of $O_K$ be integrally closed?
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40 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: ...
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60 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
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62 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 non-...
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24 views

Uniquely determined discrete valuations

Let $K$ be a nonarchimedian discrete valued field. Let $f$ be a monic irreducible polynomial in $K[x]$. Let $w$ be an extended valuation to the splitting field of $f$. (The values of roots of $f$ are ...
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49 views

Divisibility in the ring of integers.

For example, let $R=\Bbb Z [\sqrt{-5}]$, and I want to explain $3$ does not divide $2-\sqrt{-5}$. I think the following proof will be right: Suppose $3(a+b\sqrt{-5})=2-\sqrt{-5}$, then taking ...
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70 views

Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
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46 views

What does “A mod P generates the residue class field extension” mean?

We have K and finite algebraic extension L. P is a prime ideal in $O_{L}$ over prime $p\in O_{K}$ and $A\in O_{L}$. Then the problem says $\bar{A}:=A ~mod ~P$ generates the residue class field ...
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108 views

Roots of unity in a residue field of a Cyclotomic extension

Neukirch makes the following assertion in Algebraic Number Theory: Let $L = \mathbb{Q}(\zeta)$ where $\zeta$ is a primitve $n$th root of unity. Let $p$ be an integer coprime to $n$. For any prime ...
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Methods for finding Number of roots in $\mathbb{Q}_{p}$ for a polynomial

The problem is: find how many roots in $Q_{p}$ does $x^{3}+25x^{2}+x-9$ have for p=2,3,5,7. I found for $\mathbb{Z}_{p}$. Is there a way to extend from here? Also, can you suggest me a book or link ...
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75 views

Factorization into prime ideals

I citate Serge Lang's Algebra (Second edition, page 23). If $A$ is a discrete valuation ring, then in particular, $A$ is a principal ideal ring, and any finitely generated torsion-free module $M$ ...
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122 views

When does coprimality carry over to the base ring in an integral extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
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78 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} \...
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128 views

Density of set of splitting primes

Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$. Then the claim is that the ...
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59 views

Quadratic Reciprocity as a consequence of Eisenstein Reciprocity

I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how ...
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38 views

Using discriminants to find order of extension

Any hints on how to show $[G:H]^{2}=\frac{disc(H)}{disc(G)}$,where G,H are free abelian groups of rank n and $H\subset G\subset K$,where K is a number field? Alternative formulation, how to relate $[...
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63 views

Incomplete Proof from Neukirch about modules and algebraic integers

There is a theorem in Neukirch which says If $L | K$ is separable and $A$ is a principle ideal domain, then every finitely generated $B$-submodule $M \neq 0$ of $L$ is a free $A$-module of rank $...
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126 views

Being Galois stable under completion?

Let $R$ a Dedekind Domain, $K = \mathrm{Frac}(R)$ the fraction field, $L/K$ a finite galois extension, $R'$ the integral closure of $R$ in $L$. Then $R'$ is Dedekind again. Let $\mathfrak{p} \subset ...
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187 views

Field extension trace/norm confusion pertaining to multiplication matrix

I've gotten stuck on page 37 on P Samuel's 'Algebraic Theory of Numbers', on an equation that also features at the start of chapter 12 of Ireland and Rosen. The setup is, if we have a field $K$, and $...
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91 views

Questions about a proof in Greenberg's Book.

I am trying to understand the proof of the following lemma : Lemma ' : Suppose that $X$ is a finitely generated $\Lambda$-module ($\Lambda =\mathbb Z_p[[T]]$) and that $f(T)=\sum_{i=0}^{+\infty}...
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89 views

The order of the cokernel of an endomorphism over $ \mathbb Z_p$

I want to prove the following result : Let $X$ a finite-rank free $\mathbb{Z}_p$-module, and $\varphi \colon X \to X$ an endomorphism of $X$. Then $$|M/\varphi(X)| < \infty \Leftrightarrow v_p(\...
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159 views

How to show 2 cannot be totally ramified in $\mathbb{Q}(\sqrt{6},\sqrt{10})$?

I am trying to show that 2 cannot be totally ramified in $\mathbb{Q}(\sqrt{6},\sqrt{10})$. I know that it is totally ramified in $\mathbb{Q}(\sqrt{6})$ and $\mathbb{Q}(\sqrt{10})$ since $2O_{\mathbb{Q}...
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75 views

About Local fields

Let $\widehat{L}/\widehat{K}$ be an extension of local field we know there are a number field $L$ and a place $\frak P$ of $L$ such that $\widehat{L}=L_{\frak P}.$ 1) How we can prove that $\widehat{...
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164 views

Ramification of a Galois extension

I understand that an extension of number field $L/K$ is unramified if every non-zero prime ideal of $\mathcal{O}_K$ is unramified in $L$ (where a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ is ...
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136 views

Finite and infinite primes of a number field

Let $K$ be a number field. How to find the finite primes of a number field. can any one give some illustrations.
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71 views

Computing ring class field from ray class field

Let $K=\mathbb Q(\sqrt{-d})$ be any imaginary quadratic field. Let $O_{\mathcal{K}}$ be its maximal order and $O$ be any order. Let $m$ be the conductor of $K$. Is it possible to compute ring class ...
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100 views

Rings with finitely many prime ideals.

What follows is an argument i found in my textbook which i can't understand. Let $R$ be a Dedekind domain, with quotient field $K$, $R'$ is the integral closure of $R$ in a finite and separable ...
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156 views

Isomorphism of the ideal class group with a cyclic group

Let $K=\mathbb{Q}(\sqrt{-17})$. Show that the ideal class group $Cl_K$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$. We know that the class number is 4...How i show that $Cl_K$ is cyclic?
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Ireland-Rosen Hecke Character for $y^2=x^3-Dx$

I would like to refer you to page $310$ of Ireland-Rosen: A Classical Intro to Modern Number Theory. Firstly, to construct the Hecke character, it is enough to specify $\chi(P)$ for prime ideals $P$ ...
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Quadratic equation family with largest real root in Cyclotomic extension

Let the $\alpha_{k}$ be the largest real root by absolute value of $ 2x^2-2kx-(k-1)=0$ for all $k\ge1$. For what values of $k$ does $\alpha_{k}$ sit in a cyclotomic extension? How does one explicitly ...
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111 views

Factoring Monic Polynomials in $\mathbb{Z}[x]$ over $\mathbb{Z}_p$

Suppose that $f \in \mathbb{Z}[x]$ is irreducible. Then $f$ factors as $\prod\limits_{i=1}^n {f_i}^{e_i}$ over $\mathbb{Z}/p \mathbb{Z}$ and by Hensel's Lemma, each term ${f_i}^{e_i}$ lifts to some ...
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What is known on bounds comparing $ [\mathcal{O}_K :\mathbb{Z}[\alpha ]] $ to $\text{disc}(K)$, $\alpha$ a unit?

Let $K = \mathbb{Q}(\sqrt{41})$, $\omega = \frac{1}{2}(1+\sqrt{41})$, and $\mathcal{O}_K= \mathbb{Z}[\omega ]$ be the ring of integers of $K$. Let $\alpha = 27 +10 \omega$ be the fundamental unit of $\...
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Definition: Gauss Sum - Where is the error?

In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct. My intuition says "there is a mistake somewhere". I tried double-...
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130 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, ...
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188 views

Relative trace and algebraic integers

In a number field, the trace of an element over $\mathbb{Q}$ gives necessary conditions on the algebraic integers--the trace of an algebraic integer over $\mathbb{Q}$ is an integer. But when the ...
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89 views

Show that the $\mathbb{Z}$-span $\mathbb{Z}b'_1+\cdots+ \mathbb{Z}b'_d$ of $B^+$ does not depend on the choice of $B$

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that $\mathcal{O}_K = \mathbb{Z}b_1+\cdots+\mathbb{...
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83 views

Finding $n$th powers in an Integral Domain

I am part of a small group of math majors from the University of Maryland. We did NOT pursue careers in mainstream mathematical areas and now closing in on retirement we have formed a small study ...
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91 views

Is the Idele class group Hausdorff?

I was wondering if for a global field (function or number field) $K$, is $C_K$ Hausdorff? Thank you