Questions related to the algebraic structure of algebraic integers

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Question regarding to Dedekind domain and PID

I just wondering if the following statement is true. If $R$ is a Dedekind domain and $P$ is a prime ideal of $R$, then $R_P$ is a PID. $R_P$ means $R$ localize at $P$. Thanks.
2
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1answer
206 views

Conjugacy classes in GL(n)

Given an element $\gamma$ in $GL(n,F)$, where $F$ is either a global field or a non archimedean local field. Assume $\gamma$ is elliptic, i.e. its characteristic polynomial irreducible. Let $Z(F)$ be ...
3
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3answers
203 views

Effective equality

(I believe this belongs to computational algebraic number theory, but if additional/different tags are better, let me know.) Consider the cubic equation $$x^3-3x^2+x-3=0.$$ We have that $x=3$ is a ...
21
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1answer
650 views

Ring of integers in p-adic field

How do we compute the ring of integers in a finite extension of $\mathbb{Q}_p$? Say, for example, in $\mathbb{Q}_p(i)$. Over $\mathbb{Q}$ we would guess $\mathbb{Z}[i]$, compute the discriminant of ...
7
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3answers
520 views

Is there any trivial reason for $2$ is irreducible in $\mathbb{Z}[\omega],\omega=e^{\frac{2\pi i}{23}}$?

This naive question came as the last problem in my homework. The author asked me to use linear relations of the discriminant like ...
2
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3answers
359 views

What condition that if imposed on $\alpha$ imply that $\cos^{-1} \alpha$ is a rational multiple of $\pi$?

It is well known that if $x$ is a rational multiple of $\pi$ then $\cos x$, $\sin x$, etc, are algebraic numbers. What is known about the inverse problem? That is, is there a set of conditions that ...
2
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0answers
57 views

Squares in residue characteristic two

Let $F$ be a Henselian field. How can I compute the algebraic structure of $F^\times / (F^\times)^n$, if the residue characteristic divides $n$?
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1answer
375 views

How many square numbers are there?

Let $F$ be a non-archimedean local field or a general global field. Has $F^\times / (F^\times)^2$ cardinality $2$?
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1answer
149 views

$\mathbb Z_p$-rank $\leq r_1+r_2-1$ in Leopoldt's conjecture

I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields". For an algebraic number field $K$ let $E$ denote the global units, ...
9
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1answer
1k views

Is $\mathbb Z[\sqrt{-3}]$ Euclidean under some other norm?

I know that $\mathbb Z[\sqrt{-3}]$ is not a Euclidean domain under the usual norm $N(x + y\sqrt{-3}) = x^2 + 3y^2$, but that does not necessarily mean that it can't be a Euclidean domain. Is it ...
2
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1answer
227 views

How many solutions for $x^2 = 1$?

Let $F$ be an non-archimedean local field, let $o$ be its ring of integers, and let $p$ be the maximal ideal Is there a closed form for the cardinality $$ | \{ x \in o / p^N: x^2 = -1 \bmod p^N\} | ...
3
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1answer
151 views

Solving an equation in a cyclotomic field

Let $\zeta = e^{2\pi i/110}$, and set $K = \mathbb{Q}(\zeta)$. There is an $\alpha$ in $\mathbb{Q}(\zeta^{11})$ of absolute value 1, which I'm trying to find. Consider $\sigma$, the Galois ...
4
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0answers
86 views

Salem numbers and $x^{12}-x^7-x^6-x^5+1$

Given, $x^{12}-x^7-x^6-x^5+1 = 0$ This has Lehmer’s decic polynomial as a factor, hence one of its roots is the smallest known Salem number. All twelve roots obey the beautiful cyclotomic relation, ...
3
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1answer
249 views

Prove that $\mathbb{Q} \left( \sqrt[n]{p}\right) \neq \mathbb{Q} \left( \sqrt[n]{q}\right)$.

Given two distinct prime numbers $p$ and $q$, how can we prove that $\mathbb{Q} \left( \sqrt[n]{p}\right) \neq \mathbb{Q} \left( \sqrt[n]{q}\right)$ where $\sqrt[n]{p}$,$\sqrt[n]{q}\in \mathbb{R}$ and ...
2
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3answers
377 views

Calculating quotient rings

How do I go about calculating what $R = \mathbb Z [\sqrt{-5} ] / \langle 1 + \sqrt{-5} \rangle $ actually is (i.e. how do I find a simpler ring isomorphic to $R$)? I can see that $\langle \sqrt{-5} ...
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4answers
184 views

Are fractional ideals usual ideals?

I'm going through some notes, and have the following definition: Let $K$ be a number field. Then $ \mathfrak{a} \subset K$ is a fractional ideal if there exists a non-zero $c \in K$ such that ...
7
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3answers
328 views

$-1$ as the sum of three squares of algebraic integers in $\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}$

Following the result of Ivan Niven, in his paper http://www.ams.org/journals/tran/1940-048-03/S0002-9947-1940-0003000-5/S0002-9947-1940-0003000-5.pdf That is whenever $d\equiv 3 \pmod 4$, every ...
4
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1answer
464 views

A sufficient condition for a domain to be Dedekind?

We know that in a Dedekind domain, every nonzero ideal admits a unique factorization into a product of prime ideals. I was wondering if this condition is sufficient for a domain to be Dedekind, ...
7
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1answer
365 views

Special privilege enjoyed by Elliptic Curves with Complex Multiplication

I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication. I have been reading many theorems, ( ...
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5answers
255 views

Infinite distinct factorizations into irreducibles for an element

Consider the factorization into irreducibles of $6$ in $\mathbb{Z}[\sqrt{-5}]$. We have $6=2 \times 3$ and $6=(1+\sqrt{-5}) \times (1-\sqrt{-5})$, i.e. $2$ distinct factorizations. And, $$6^2=3 ...
3
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1answer
125 views

Computing the free-part

I wanted to ask about some existing algorithms for computing points over elliptic curves. Background : We know that the famous theorem of Mordell and Weil says that " Group of rational points on an ...
4
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2answers
224 views

Example request: unramified field extension with a relative power integral basis?

A field extension $L/K$ (of number fields) admits a relative power integral basis if $\mathcal{O}_L = \mathcal{O}_K[\alpha]$ for some $\alpha \in \mathcal{O}_L$. I'm looking for a simple example in ...
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2answers
189 views

Example demonstrating that $R=\{a+bi\sqrt5: a,b \in \mathbb{Z}\}$ is not a Euclidean domain.

We know $R=\{a+bi\sqrt{5}: a,b \in \mathbb{Z}\}$ is not a UFD because, for example, you can factor $$6=(1+i\sqrt{5})(1-i\sqrt{5})=(2)(3)$$ and these are two distinct factorizations into ...
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1answer
105 views

Generalizations of Monogenic Fields

We recall that a number field $K$ is called monogenic if there exists some $\alpha \in \mathcal{O}_K$ such that $\mathcal{O}_K = \mathbb{Z}[\alpha]$. If instead we take a tower of finite extensions ...
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1answer
181 views

Why $\sqrt{p}$ is in $\mathbb{Q}(\omega)$ when $p\equiv 1$(mod 4)? [duplicate]

Possible Duplicate: Unique quad. subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, is $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$ This question may be ...
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3answers
407 views

Every finite field is a residue field of a number field

I have read that for any finite field $\mathbb{F}_q$ there exists a number field $F$, and some prime ideal $P$ of $\mathcal{O}_F$ such that $$\mathbb{F}_q \cong \mathcal{O}_F / P.$$ The ring of ...
1
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1answer
107 views

What is the relationship between the norms $[\mathcal{O}_L : \mathfrak{a}_L]$ and $[\mathcal{O}_K : \mathfrak{a}_K]$?

Let $K$ be an algebraic extension of the rational numbers and $L$ an algebraic extension of $K$. Let $\mathfrak{a}_K = (a, \alpha )$ be an ideal of the ring of integers $\mathcal{O}_K$ of $K$, with $a ...
7
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1answer
342 views

How does the Artin symbol generalize Legendre and Hilbert symbols?

I am currently going through a course in Class Field theory and have read that the Artin symbol generalizes the Legendre and Hilbert symbols. I was wondering what field extensions to consider to see ...
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0answers
389 views

Square-free discriminants and integral bases

Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then ...
2
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2answers
333 views

How to see why $\Delta(x_1', \ldots , x_n ') = (\det A)^2 \Delta(x_1, \ldots ,x_n)$

Let $K$ be a number field of degree $n$, $\sigma_1 , \ldots , \sigma_n$ be the distinct embeddings of $K$ into $\mathbb C$, and define $\Delta(x_1, \ldots , x_n) = \det(\sigma_i (x_j)^2) = ...
6
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1answer
430 views

Algorithm for finding the discriminant of algebraic number fields

I am reading J.S. Milne's Algebraic Number Theory notes, http://jmilne.org/math/CourseNotes/ANT.pdf. I am quite confused with the section "Algorithm for finding the ring of integers". There is an ...
4
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1answer
269 views

Problems about consecutive semiprimes

I was playing around with semi-prime numbers and I made two conjectures, which are: Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime. There are infinitely many ...
2
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2answers
188 views

Wikipedia plot of $\deg(\mathrm{minpoly})$ of complex numbers?

Regarding the following picture on the Wikipedia article for Algebraic numbers: The description is: Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours ...
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1answer
89 views

$\mathcal{O}_K.\mathcal{O}_L=\mathcal{O}_{KL}$ The question is give a proof of counterexample.

I know that I am looking for a counterexample, the statement is not true when the Disc(K) and the Disc(L) are not coprime. I have been trying to use $Disc(K)=[\mathcal{O}_K:K]^2Disc(\mathcal{O}_K)$, ...
23
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2answers
435 views

Are irrational numbers order-isomorphic to real transcendental numbers?

I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers? I know that the order type of ...
3
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1answer
130 views

How to check whether these elements of a cubic number field are relatively prime?

Let $a,b$ be odd positive integers, and let $x$ be an integer not divisible by $7$ or $13$. In the field $K = \mathbb{Q(\theta)}$ with $\theta^3 = 7 \cdot 13^2$, we have $$ x^3 - 7 \cdot 13^2 \cdot ...
9
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2answers
261 views

Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$

On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with ...
2
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2answers
370 views

Computing trace and norm in a number field

Let $K = \mathbb Q(\theta)$, where $\theta$ is a root of the polynomial $f = X^3 - 2X + 6$. Then $f$ is irreducible over $\mathbb Q$, so $[K:\mathbb Q] = 3$. I'm trying to compute $N_{K/\mathbb Q} ...
3
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2answers
707 views

Method for determining irreducibles and factorising in $\mathbb Z[\sqrt{d}]$

I know that $\mathbb Z[\sqrt{7}]$ is a UFD, and I can write the equation $(2 + \sqrt{7})(3 - 2\sqrt{7}) = (5 - 2\sqrt{7})(18 + 7\sqrt{7}) $. So clearly these are not all irreducibles. How do I ...
5
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1answer
127 views

Three maximal ideals lying over $3\mathbb{Z}$?

A few weeks ago I asked a question about finding the number of maximal ideals lying above $3\mathbb{Z}$ in $B$, where $B$ is the integral closure of $\mathbb{Z}$ in a splitting extension ...
5
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1answer
154 views

What's the best way to detect an algebraic number?

Suppose you calculate the first few (dozen, hundred) digits of a number which you believe to be a rational number. You can calculate the continued fraction for the number and truncate after a large ...
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1answer
418 views

How does topology enter Number theory and how we can grasp its essence?

In infinite Galois theory,main theorem failed and we get a "Krull topology" to mend the main theorem, we even generalized that to make definition for profinite group. In local class field theory, the ...
3
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1answer
151 views

'Divides means contains' for Dedekind domains, by treating PIDs and localising

I'm trying to solve the following problem: Suppose $R$ is a Dedekind domain which contains (nonzero) ideals $\mathfrak{a}$ and $\mathfrak{b}$. By first dealing with the case where $R$ is a PID and ...
3
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1answer
646 views

split in cyclotomic field

$K=Q(\zeta_n)$ a cyclotomic extension: $p$ splits completely in $K$ if and only if $p\equiv 1\ (mod\ n)$ I don't know how i could prove, I search a kind of cyclotomic reciprocity law Many thanks
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0answers
77 views

Value groups of an archimedean field and its completion coincide

I am aware that for a non-archimedean field $K$ and its completion $\hat{K}$ with respect to a valuation $v$ (and corresponding absolute value $|\cdot|$), with $v$ extending to valuation $\hat{v}$ on ...
3
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1answer
179 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
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1answer
182 views

Discrete valuation ring extension such that $A[\pi]$ is not integrally closed

Let $B/A$ be a finite integral extension of discrete valuation rings of characteristic zero. Question. Is there a uniformizer $\pi$ in $B$ such that $A[\pi]$ is a discrete valuation ring? If not, ...
5
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1answer
219 views

Can this be salvaged to give a proof that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}(\sqrt[3]{2})$?

Recently, I was intrigued by the question asking for an easy way to show $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}(\sqrt[3]{2})$. I was playing with the approach, trying to ...
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1answer
226 views

Reference for proof of “Dedekind's Criterion”?

It was mentioned to me briefly in passing about a criterion for rings of integers, referred to as Dedekind's Criterion. The Criterion essentially said that a ring $\mathbb{Z}[\omega]$ (for ...
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1answer
113 views

the minimum polynomial of a unit

Let $A$ be a dvr of characteristic zero. Let $B/A$ be a finite integral extension of $A$. Suppose that there exists a unit $x$ in $B$ such that $B=A[x]$. What can we say about the minimal polynomial ...