All Questions
Tagged with ra.rings-and-algebras algebraic-number-theory
48 questions
15
votes
2
answers
3k
views
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
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
7
votes
1
answer
550
views
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
7
votes
1
answer
633
views
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
7
votes
0
answers
92
views
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
6
votes
1
answer
290
views
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
6
votes
0
answers
204
views
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
6
votes
0
answers
143
views
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
5
votes
3
answers
550
views
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
5
votes
1
answer
271
views
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
5
votes
1
answer
335
views
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 ...
5
votes
0
answers
181
views
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
4
votes
1
answer
266
views
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} ...
4
votes
2
answers
229
views
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
4
votes
1
answer
173
views
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
4
votes
0
answers
164
views
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
0
answers
85
views
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
3
votes
1
answer
134
views
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
3
votes
1
answer
180
views
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
3
votes
1
answer
114
views
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
3
votes
0
answers
150
views
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
3
votes
0
answers
79
views
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
votes
2
answers
384
views
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
2
votes
2
answers
416
views
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
views
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
2
votes
1
answer
146
views
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
2
votes
1
answer
745
views
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
1
answer
216
views
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{...
2
votes
1
answer
216
views
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
2
votes
0
answers
119
views
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
0
answers
147
views
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 $\...
2
votes
0
answers
48
views
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
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
2
votes
0
answers
109
views
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
1
vote
2
answers
685
views
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
1
vote
1
answer
488
views
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" ...
1
vote
1
answer
80
views
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
1
vote
1
answer
142
views
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
1
vote
0
answers
118
views
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
1
vote
0
answers
200
views
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
1
vote
0
answers
63
views
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
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
1
vote
0
answers
189
views
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
1
vote
0
answers
62
views
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
0
votes
2
answers
369
views
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
0
votes
0
answers
108
views
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
0
votes
0
answers
186
views
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