Questions related to the algebraic structure of algebraic integers

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132 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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67 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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154 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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76 views

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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89 views

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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110 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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117 views

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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108 views

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 ...
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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, ...
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186 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 = ...
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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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1answer
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
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73 views

Number of Q-embeddings into C

Let $\alpha $ be a root of the polynomial ${x^5} + 6{x^3} + 8x + 10$. How many $\mathbb{Q}$-embeddings of $\mathbb{Q}\left[ {\alpha ,\sqrt 7 } \right]$ (the least field extension of $\mathbb{Q}$ which ...
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87 views

Selfconjugate prime ideal of a cyclic extension of an algebraic number field of prime degree.

Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $L$ be a cyclic extension of $K$ of degree $l$, where $l$ is a prime number. Let $B$ be the ring of integers in $L$. ...
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72 views

Normality of a certain localization 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 $p$ be a prime number. ...
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302 views

An exact sequence on the ideal class group of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. It is well-known ...
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146 views

Finding the matrix of multiplication by $\theta^2$, where $\theta^3 - 3\theta + 1 = 0$

This is a problem from a on-line source which yet comes with a solution (self-studier; not h.w.). Let $E = \mathbb Q(\theta)$, where $\theta$ is a root of the irreducible polynomial \[ X^3 -3X + 1. ...
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322 views

A lemma on the integral closure of a Noetherian domain of dimension 1

I need to prove the following lemma(?) which is motivated by this and this. Lemma Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral ...
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767 views

$\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ ...
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162 views

Splitting of primes in the splitting field of a polynomial

Let $K$ be a number field and $f(X)\in K[X]$. Let $E$ be the splitting field of $K$, so that we know that the set of primes splitting in $E$ has density $1/[E:K]$. Milne uses this as the argument ...
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206 views

Tamagawa number conjecture

I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please ...
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223 views

Decomposition of prime ideals

First of all, the polynomial $p(x)=x^{3}-x-4$ must have one real root. Let $\theta$ be one of its roots, L=Q($\theta$), $O$ the integral closure of Z in L. Then Kummer's theorem tells us then and ...
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Confusion about definition of primitive polynomials

I am working through Neukirch's Algebraic Number Theory and am confused about his definition of primitive polynomials on page 129. He defines $f(x)=a_0+a_1x+\dots +a_nx^n$ on $\mathcal{O}$ with ...
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50 views

Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is ...
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29 views

Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
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64 views

Norm and trace of $\sqrt{15}$ over $K = \Bbb Q(\sqrt3, \sqrt5)$

I have been stuck on an algebraic number theory question, could you please show me how you would approach this: work out the norm and trace of $\sqrt{15}$ over the number field $K = \Bbb Q(\sqrt3, ...
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40 views

Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
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29 views

Results in algebraic number theory regarding ramified split and inert primes in quadratic fields

I am currently reading some notes in algebraic number theory but they are not really self contained and I am guessing the following results must hold. Let $K$ be a quadratic field and consider the ...
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22 views

Algebraic number with conjugates having modulus 1

Suppose $\alpha$ is an algebraic number lying in a number field $K$ that is a normal extension of $\mathbb{Q}$. Suppose all the conjugates of $\alpha$ have absolute value 1. Prove or disprove that it ...
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58 views

Absolute Galois group of quadratric extensions

It is well known that the absolute Galois group of $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{q})$ are nonisomorphic if $p$ and $q$ are different prime numbers. See for example Szamuely's book ...
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42 views

Problem from the book Number Fields by Marcus

I have been stuck on the 14(c)th problem of the 3rd chapter from Marcus' Number Fields. Let $K$ and $L$ be number fields, $K \subset L$, $R = \mathbb{A}\cap K$, $S = \mathbb{A} \cap L$. Moreover ...
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21 views

Basis of neighbourhoods in a profinite group

The Krull topology in a Galois group $G$ of a Galois extension $L/K$ is defined taking $\sigma\:G(L/M)$, where $M/K$ varies through the Galois finite subextensions of $L/K$, as a fundamental system of ...
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30 views

Showing $A[\theta] \subseteq B \subseteq \frac{1}{d} A[\theta]$, where $A$ is a Dedekind domain

Suppose we have $A$ a Dedekind domain, $K$ its field of fractions. Let $L$ be a field extension of $K$ of degree $n$ and also that it is separable. Let $B$ be the integral closure of $A$ in $L$. ...
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31 views

Is the set of real algebraic numbers in $(0,1)$ the same as the set of fractional parts of real algebraic numbers in $(1, \infty )$?

It seems that way to me, but I'm not sure how to prove it rigorously. Say, we have the number $x>1$ that is a root of some polynomial with integer coefficients: $$a_0+a_1 x+a_2 x^2+\dots +a_n ...
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54 views

What are the invariants of a number field? [closed]

How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated ...
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21 views

Bounds for zeta function residue

Let $K$ be an algebraic number field and let $c = c(K)$ denote the residue at $s = 1$ of its zeta function. It is known Wikipedia: class number formula that c can be determined via $$c = \frac{2^r ...
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33 views

Completion of a Galois Extension $K | \mathbb{Q}$ w.r.t. to a non archimedean abs. value is a Galois extension

As the title suggests, I'm interested in a proof of the following fact: Let $K | \mathbb{Q}$ be a finite Galois extension, and let $\mathfrak{p}$ be a prime ideal of the ring of integers of $K$. ...
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1answer
34 views

Is this ring a Dedekind domain?

Let $p(x)\in \mathbb{Z}[x]$ be a monic, irreducible polynomial in $\mathbb{Z}[x]$. For the field $K:=\mathbb{Q}[x]/(p(x))$, its ring of algebraic integers $\mathcal{O}_K$ is always a Dedekind domain ...
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58 views

Local reciprocity map applied to norm

This question concerns the local reciprocity homomorphism $r_L : L^\times \to G_L^\text{ab}$, where $L$ is a local field with absolute Galois group $G_L$. If $K\subset L$ is a subfield and $r_K$ is ...
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60 views

What are the pre-requisites required to understand Milnor's book on algebraic K- theory?

I want to understand Steinitz’ theorem on the structure of finitely generated modules over Dedekind domains. I also want to have some general awareness regarding what Algebraic K-theory is about. ...
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23 views

Factorization of extension is injective

Let A be a Dedekind domain with field of fractions K, and let B be the integral closure of A in a finite separable extension L of K. Now I want to show the map from Id(A) to Id(B) is injective. I know ...
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44 views

$\mathcal{O}_L$ free over $\mathcal{O}_K[G]$

Let $L/K$ be a galois extension of number fields. Suppose $G:=\text{Gal}(L/K)$ is abelian. If $\mathcal{O}_L$ is free as $\mathcal{O}_K[G]$-module, is it true that it has rank 1?
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63 views

Proof of Proposition 2.12 in Neukirch ANT

I'd like a reference or a direct proof of the following statement: Let $K|\mathbb Q$ be a finite extension and consider the ring of algebraic integers $\mathcal O_K$. Let $\mathfrak ...
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1answer
49 views

Find polynomial to use for prime ideal factorization

L.S., In an exercise for my algebraic number theory homework I came across the following problem: I would like to factor ideals $(2)$ and $(7)$ in $K = \mathbb{Q}(i, \sqrt{14})$. I managed to show ...
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21 views

Is $\widehat{K}L$ complete?

Let $K$ be a field and $\widehat{K}$ be a completion with respect to some valuation on $K$. Let $L$ be a finite separable extension of $K$. When regarded as a subfield of $\widehat{L}$, is ...
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62 views

Number field extension

Given a number field K show that there exists a number field extension L of K such that every ideal in K becomes a principal ideal in L.
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33 views

factor ideal $(3)$ in biquadratic field

L.S., I would like to factor $(3)$ in $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of $f = X^4 + 4X^2 + 2$. I need this factoring as a part of an exercise I need to do from my course on ...
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48 views

prime ideals in extensions of dedekind domains

Let $A$ be a Dedekind domain and $B$ its integral closure on a finite extension of its field of fractions. Is it true that if $\mathfrak{I}$ is an ideal of $B$, then factorizing it as ...