Questions related to the algebraic structure of algebraic integers

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Is there a redundant assumption in Exercise 2, page 14, from Janusz “Algebraic Number Fields”?

The exercise says: Let $R$ be an integral domain with quotient field $K$ and let $M$ be an $R$-submodule of a finite dimensional $K$-vector space. Prove $M=\bigcap_{P} R_P M$, where the ...
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36 views

construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$?

how might I construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$? Can it be done using pigeonhole principle as with square roots and Pell equation. I had been reading about the Voronoi continued fraction or ...
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38 views

Nice shapes of ideals of $\mathbb{Z}[i]$ from a (lattice) geometric point of view?

If we draw the lattice for the ideal generated by $(2+i)$ in $\mathbb{Z}[i]$, and look at what is happening modulo $(2+i)$, we see a beautiful square, although it is rotated a little bit counterclock-...
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59 views

Questions that SAGE, MAGMA can answer?

I practice theoretical mathematics and I know (almost) nothing about SAGE, MAGMA. I would like to know (in general) what type of questions can I ask SAGE to do? For example, I know that given an ...
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42 views

Why assuming that the ideal in Minkowski's bound is prime

Minkowski’s bound states that given a quadratic field $K(\sqrt{d})$ then every class of ideals in $\mathcal{O}_K$ contains an integral ideal of norm<$\lambda(d)$. Then my notes say that this ...
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68 views

Galois group is isomorphic to group of characters. What can we say then?

Let $\overline{K}/K$ denote the separable closure of a finite field $K$ of characteristic $p$ and let $\mu_{n}$ denote the group of $n$-th roots of unity in $\overline{K}$, where $(n,p)=1$. Let us ...
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Showing $B_P$ is a finitely generated module over $A_P$ where $P$ is a prime ideal in 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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Ramification of primes under cetain conditions

Let $K\subset L=K(\gamma)$, ($\gamma$ an algebraic integer) be number fields such that there exists a $k\in \mathcal O_K$ and some $n\in\Bbb N$, $\gamma^n=k$. Also, there exists an ideal $I\subset \...
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35 views

eigenvalues of sum of matrices with algebraic integers eigenvalues

Let $A, B$ be two matrices such that they both have all eigenvalues in $\mathbb{A}$, the ring of algebraic integers. The question is: it is true that the matrix $A+B$ does have all of its eigenvalues ...
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56 views

On the genus of a curve and its set of rational points.

The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for ...
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Prove a property of the order of conductor $f$ in the field $\mathbb{Q}(\sqrt{D})$

Let $D$ be a squarefree integer, and let $\mathcal{O}$ be the ring of integers in the quadratic field $\mathbb{Q}(\sqrt{D})$. For positive integer $f$ define the order of conductor $f$, $\mathcal{O}...
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42 views

Proof that the Galois Field of order 8 is a field.

We know that the Galois Field of order 8 is isomorphic with $$\left( Z_2[x]^{< 3}, +_{d(x)}, \times_{d(x)}\right) $$ (Field of polynomials with coefficients in $Z_2$ and of grade smaller than 3, ...
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141 views

Can there be a set of numbers, which have properties like those of quaternions, but of dimension 3? [duplicate]

We all know what the complex numbers are : basically $$R^2$$ with a specified product formula, which is $$(a,b).(c,d)=(ac-bd,ad+bc).$$Hamilton defined the quaternions, which are a wonderful set of ...
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52 views

No ideals coprime in DVR?

There's this exercise in Neukirch, chapter I, §3 (i've changed the statement to deal only with the case that bothers me): Let $\mathfrak o$ be a Dedekind domain and $\mathfrak m$ be a nonzero ...
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39 views

When is norm surjective mod an ideal for global fields?

Given $K/k$ Abelian, for which ideals of $\frak p\unlhd\cal O_k$ will we have $N^K_k:\cal O_K\rightarrow\cal O_k/\frak p$ surjective? $k$ is an algebraic number field. In his article "On the norm ...
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Relation between a maximal ideal and an invertible ideal

Let $D$ be a domain, $\mathfrak{a,b,p}\subsetneq D$ ideals with $\mathfrak{p}$ maximal and $\mathfrak{a}$ invertible (there is some $\mathfrak{c}$ ideal with $\mathfrak{ac}=\mu D$, with $\mu\in D\...
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51 views

Question regarding the definition of different of a number field

Let $K$ be a number field. I was getting a bit confused, because different sources I looked into had different definitions. I was wondering if they were equivalent or not. And if they are how can I ...
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41 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 ...
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Does this matrix involving roots of unity has a particular name?

Do the matrices of the form $\left(\begin{matrix} \frac{\xi + \xi^{-1}}{2} & \frac{\xi - \xi^{-1}}{2} \\ \frac{\xi - \xi^{-1}}{2} & \frac{\xi + \xi^{-1}}{2} \end{matrix}\right)$ where $\xi$ is ...
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How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
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67 views

Ramification in $\mathbb Q(\zeta_5, \sqrt[5]2)/\mathbb Q(\zeta_5)$

Let $F=\mathbb Q(\zeta_5,\sqrt[5]2)$ and $K=\mathbb Q$ where $\zeta_5$ is a primitive $5$th root of unity and let $p=73$ be a prime in $K$. Fix primes $\mathfrak p$ and $\mathfrak q$ above $73$ in $\...
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Why $\mathfrak p_2\cdots\mathfrak p_r\not\subset (a)\mathcal O$

If $(a)\mathcal O\subset\mathfrak p_1$ and $r$ is the minimal number such that $\mathfrak p_1\cdot\mathfrak p_2\cdots\mathfrak p_r\subset (a)\mathcal O$ then $\mathfrak p_2\cdots\mathfrak p_r\not\...
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Is it possible to factor $27$ in $\mathbb Z[\sqrt7]$ other than in $27=3\cdot3\cdot 3$?

I need to know if there exists an element of norm$27$ in $\mathbb Z[\sqrt 7]$, that is, whether there are any integer solutions to $a^2 - 7b^2 = 27$. When I used modular arithmetic, I find out that $a^...
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158 views

Computing class group of $\mathbb Q(\sqrt{6})$

I am calculating the class group of $\mathbb Q(\sqrt 6)$. My working is as follows: The Minkowski bound is $\lambda(6)=\sqrt 6<3$ so we only need to look at prime ideals of norm $2$. $2$ divides ...
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75 views

Quotient of the ring of integers of a quadratic field by the ideal generated by a split integer prime.

I am wondering about primes $p$ in $\mathbb Z$ that are split in $\mathcal O_{K}$, $K=\mathbb Q(\sqrt d)$. Let $\omega=\sqrt d$ if $d \equiv 2,3 \mod 4$ and $\omega=\frac{1+\sqrt d}{2}$ if $d \equiv 1 ...
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60 views

Extension of Completions of Number Fields

On p. 116 of Milne's notes on Algebraic Number Theory, he gives the following construction. Let $K$ be a field with a valuation $|\cdot|$ (archimedean or discrete nonarchimedean), and let $L$ be a ...
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58 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is it ...
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Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
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Condition on ideal of ring of integers being prime

Let $K=\mathbb{Q}(\sqrt{d})$ for $d$ square free integer and let $p$ be a rational prime such that $p$ does not divide $2d$. Prove that $p\mathcal{O}_K$ is a prime ideal $\iff x^2\equiv_p d$ has no ...
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If the limit of a sequence of algebraic integers is algebraic, does it need to be an algebraic integer?

Consider a sequence $\{\alpha_n\}$ of algebraic integers and let $\alpha = \lim_{n \to \infty} \alpha_n$, where the limit is taken with respect to the usual absolute value in $\mathbb{C}$, and suppose ...
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Norm of an ideal is finite

I want to show that the norm $N_{K/\mathbb Q}(\mathfrak{a})$ of $\mathfrak{a}$ a nonzero integral ideal of a number field $K$ is finite, and so $N_{K/\mathbb Q}(\mathfrak{ab})=N_{K/\mathbb Q}(\...
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Ideals in the ring of integers $\mathcal{O}_K$

I have two question about the proof of this theorem: Theorem: Let $K$ be an algebraic number field and $n = \dim_{\mathbb{Q}} K$. Then any ideal $I\neq 0$ in the ring of integers $\mathcal{O}_K$ ...
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Discriminant of a Polynomial over a Local Field

I am trying to prove the local Kronecker-Weber theorem for tamely ramified abelian extensions $L|\mathbb{Q}_p$. At some point in the proof I need to show that $\mathbb{Q}_p(u^{1/e})$ is unramified ...
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184 views

Norm and Trace of an element is an integer, then element is an integral?

Let $L/K$ be a finite field extension, and let $\{b_1,b_2,...,b_d\}$ be a basis for $L/K$ My notes define $O_k:=\mathbb{B}\cap K$, where $\mathbb{B}:=\{\alpha$ is algebraic|min poly of $\alpha$ ...
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215 views

Problem solving Neukirch's Algebraic Number Theory, Exercise 1.7.4

I have tried to solve exercise 1.7.4 in Neukirch's Algebraic Number Theory which states that $1+\zeta $ is a fundamental unit of $\mathbb Z [\zeta]$ when $\zeta$ is a primitive $5$th root of unity. I ...
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Ideal as kernel of a homomorphism in Gaußian integers

Consider the ring $\mathbb{Z}[i]$ of Gaußian integers. The principal ideal $(1+i)$ is maximal ideal in this ring. Since ideals are kernels of some homomorphisms, I would like to see a homomorphism ...
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Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic $a_1 \in \mathbb{Z}_p$ such that $$F(a_1) \equiv 0 \mod p \...
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57 views

Eigenvalue of multiplication in a number field.

Let $\mathbb{K}$ be an algebraic number field. $b \in \mathbb{K}$ defines a linear transformation $\phi(b)$ on $\mathbb{K}/\mathbb{Q}$ by multiplication - $\phi(b)(x):=bx$ for all $x\in\mathbb{K}$. ...
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269 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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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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188 views

Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers

I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1): Show that, in the ring $\mathbb{Z}[i]$, the relation $\alpha\beta=\varepsilon\gamma^n$,...
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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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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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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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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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What is the class number of $\mathcal{O}_{\sqrt[3]{18}}$?

I accept it without proof that $\mathcal{O}_{\sqrt[3]{2}}$ and $\mathcal{O}_{\sqrt[3]{3}}$ both have class number $1$. Also, I've been told that $\mathcal{O}_{\sqrt[3]{m^2}} = \mathcal{O}_{\sqrt[3]{m}}...
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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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110 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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61 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 ...