All Questions
Tagged with algebraic-number-theory ra.rings-and-algebras
21 questions with no upvoted or accepted answers
7
votes
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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
6
votes
0
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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
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
5
votes
0
answers
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
4
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0
answers
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
0
answers
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
3
votes
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?
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- 26.5k
3
votes
0
answers
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
votes
0
answers
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
0
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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 $\...
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
2
votes
0
answers
142
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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
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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
1
vote
0
answers
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
1
vote
0
answers
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
1
vote
0
answers
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
1
vote
0
answers
50
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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
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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
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137
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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
0
answers
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
0
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0
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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?
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