All Questions
Tagged with algebraic-number-theory ra.rings-and-algebras
48 questions
1
vote
1
answer
80
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Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full
in order to find an explicit ...
- 155
3
votes
1
answer
114
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Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
- 319
4
votes
2
answers
229
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Arithmetic application: Complete group ring and group ring for infinite group
Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\...
- 319
2
votes
0
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119
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gcrd and associates of an element of the quaternion algebra over a totally real number field $K$
Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis
$\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
- 133
2
votes
2
answers
416
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Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$
Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...
- 319
2
votes
2
answers
432
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Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
1
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0
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118
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Degrees of trigonometric numbers
For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers.
What is its degree?
That is, what is the minimal degree of ...
- 8,086
7
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1
answer
633
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Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
5
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0
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181
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The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?
In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...
- 7,127
2
votes
2
answers
384
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A ring map from algebraic integers to algebraic closure of $\mathbb F_p$
Let $p$ be a prime and ${\mathbb F}_p$ the finite field with $p$ elements. There is a canonical ring map ${\mathbb Z} \to {\mathbb F}_p \cong {\mathbb Z}/ p {\mathbb Z}$. Denote the image of $n$ by $[...
- 803
4
votes
1
answer
266
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Positive system of algebraic integers
Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
1
vote
0
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200
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Units in residue classes modulo prime ideal
Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}...
- 11
3
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1
answer
134
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Trace-free basis for $\mathcal{O}_K$, $K$ a cubic field
Let $K$ be a cubic field and let $\mathcal{O}_K$ be its ring of integers. Does there always exist elements $\alpha, \beta \in \mathcal{O}_K$ with $\text{Tr}(\alpha) = \text{Tr}(\beta) = 0$ such that $\...
- 26.9k
5
votes
1
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335
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About the structure of unit groups appearing in number theory
I think the following statement is not true in the general situations, but consider it:
$R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...
6
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0
answers
204
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The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?
I asked this question over a year ago on Math.StackExchange but I didn't get an answer.
In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
- 841
7
votes
1
answer
550
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Explicit construction of division algebras of degree 3 over $\mathbb{Q}$
In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $\mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/...
- 767
3
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0
answers
150
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For which $n$ is this ring an euclidean domain?
Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$.
Question: Is $A_n$ for all $n$ an euclidean domain? Is there a good choice for an euclidean function?
...
- 26.5k
1
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0
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63
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Maximal orders separable over their centre
Let $\mathcal{A}$ be a central simple $K$-algebra, where $K$ is an algebraic number field. It is known that $\mathcal{A}$ is separable over $K$ (following the definition of DeMeyer and Ingraham's book)...
- 323
7
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0
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92
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Is the set of conjugates of Pisot numbers dense?
Let $S$ be the set of Pisot numbers. It is known that $S$ is closed and has infinitely many limit points. However, I want to know if there are are results about the set of conjugates of Pisot numbers. ...
- 171
3
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0
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79
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The type number of an algebra
I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
- 323
2
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1
answer
216
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A problem about an unramified prime in a Galois extension
I asked this question in MathStackExchange, but I didn't receive any answer.
Let $K/\mathbb{Q}$ be a Galois extension of degree $n$, and denote its ring of integers by $\mathcal{O}_K$. Let $\mathfrak{...
1
vote
1
answer
488
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Polynomials for the alternating group $A_n$
It is my understanding that the polynomial $f_n(x)=x^n-1$ has the Galois Group $(\mathbb{Z}/n\mathbb{Z})^*$, group of units of order $\phi(n)$. In some sense, these are the "simplest" ...
2
votes
0
answers
147
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Characterization of algebraic integers providing a prime ideal
Let $\alpha$ be an algebraic integer and let $\mathcal{O}_{\mathbb{Q}(\alpha)}$ be the ring of integers of $\mathbb{Q}(\alpha)$.
Question: How to characterize an algebraic integer $\alpha$ such that $\...
1
vote
1
answer
142
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Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request
I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres"
On the 2nd section, they recall some facts on valuations, ...
- 1,013
6
votes
0
answers
143
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Newer versions of Mahler's Lemma
I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer).
The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...
- 563
2
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1
answer
745
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Motivation to study the order theory (ring theory)
I'm currently reading a paper of Georges Gras on the Reflection Principle.
The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting,...
- 1,013
2
votes
0
answers
48
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Monogenic suborders of irreducible cubic orders
By an order of rank $n$ we mean a unital ring $R$ which is isomorphic to $\mathbb{Z}^n$. An order $R$ is irreducible if it is isomorphic to a subring of a degree $n$ number field $K$ and the fraction ...
- 26.9k
5
votes
1
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271
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Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings?
Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a ...
- 804
1
vote
2
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685
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Every ideal is principal in a certain quotient ring
Does anybody know if there is a valuation-free proof of the following fact?
Let $K$ be a finite extension of $\mathbb{Q}$. Then, for every $q\in
\mathbb{Z}$, the quotient ring $\mathcal{O}_{K}/\...
- 87
4
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0
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164
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Sextic resolvent rings of quintic rings
In Higher composition laws IV: The parametrization of quintic rings M. Bhargava gave an explicit parametrization of quintic rings by quadruples of $5\times5$ skew-symmetric matrices. His proof hinges ...
- 26.9k
4
votes
1
answer
173
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On the factorization of powers of atoms in the ring of integers of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is a non-unit element $a \in H$ that doesn't split into the product of two non-unit elements.
Given $x \in H$, we ...
- 10.5k
1
vote
0
answers
50
views
Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
- 10.5k
0
votes
0
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108
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Preimage of projection of idèles, and other usual maps
Let $K$ be a quadratic number field.
I am struggling with some "usual" maps in algebraic number theory, but with which I am not used to, confusing a lot of different settings, as idèles, ...
- 5,424
5
votes
3
answers
550
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Reference request: correspondence between central simple algebras and quadratic forms
Let $A$ be an algebra over $k$, $\operatorname{tr_A}(x, y):=\operatorname{tr}(m_{xy})$ be a trace form on $A$, and $V_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \...
- 1,980
0
votes
2
answers
369
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Counting Divisors in $\mathbb{Z}^n$
Basically, I'm looking for ways to multiply elements of $\mathbb{R}^n$ that allow me to count divisors in $\mathbb{Z}^n$.
For every positive integer $n$, I'm looking for an algebra structure on $\...
- 217
2
votes
1
answer
216
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Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme
Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ ...
- 7,609
1
vote
0
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189
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What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]
Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...
- 255
4
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0
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85
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An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$
Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.
Is ...
- 255
2
votes
1
answer
146
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Existence of class modules for finite groups
I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$:
1) $H^1(H,...
- 1,629
3
votes
1
answer
180
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Bibliography suggestion for Kummer theory
I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we ...
- 215
2
votes
0
answers
142
views
calculation in a group ring
I have some problems with the verification of the third equation in Lemma 1 in this paper.
First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ above....
- 31
1
vote
0
answers
62
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Valuations in Higher-dimensional local fields
I have the following question which I believ should be true but I would like to have a different opinion about it:
Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and $t_1,\...
- 11
1
vote
0
answers
137
views
Units in residue classes
Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field)
Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...
- 3,320
2
votes
0
answers
109
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classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant
Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem (...
- 21
15
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2
answers
3k
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Quintic polynomial solution by Jacobi Theta function.
Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...
- 1,594
6
votes
1
answer
290
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Algebraic integers in skew fields
Hi everyone,
let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a ...
- 183
0
votes
0
answers
186
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Ring of Integers as subring with most irreducibles
Let $L$ be a number field. Is it possible to define its ring of integers $R$ by saying it's the subring with (in a fuzzy sense) the "most" irreducibles?
user11863
10
votes
3
answers
1k
views
Infinite dimensional central simple algebras
When constructing the Brauer group of a field, only the finite-dimensional central simple algebras are considered (because of Artin-Wedderburn's characterization).
But what happens to the infinite-...
- 2,992