# Tagged Questions

**0**

votes

**0**answers

63 views

### approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...

**1**

vote

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106 views

### subring of the matrix algebra

Let $Mat_2(\mathbb{Z})$ be the $\mathbb{Z}$-algebra of $2\times2$ matrices with integer entries.
Let $A$ be a $\mathbb{Z}$-submodule of $Mat_2(\mathbb{Z})$ containing $\mathbb{Z}$. We want to show ...

**0**

votes

**0**answers

130 views

### How to find $n$ such that the group of units $U(\mathbb{Z}/n\mathbb{Z})$ has a given abelian subgroup?

Given an integer $n$, we can determine the structure of the multiplicative group of integers modulo $n$ ($U(\mathbb{Z}/n\mathbb{Z})$) by the factorization of $n$. Hence we can easily find all the ...

**4**

votes

**1**answer

206 views

### Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
...

**0**

votes

**0**answers

81 views

### Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.
Definition(Super-Gorenstein ideal): ...

**7**

votes

**2**answers

281 views

### Number of ways to write an integer as a product of irreducibles

Is there any way to tell the number of distinct ways to factor $a\in\mathcal{O}_k$ (up to units, of course) when $k$ is not a PID? A simple investigation in $\mathbb{Q}(\sqrt{-5})$ with integer ring ...

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votes

**0**answers

96 views

### How do I compute the Azumaya locus?

I have an algebra $A$ that is finite over its centre $Z(A)$ and I want to compute the Azumaya locus of $Z(A)$ (or equivalently, its ramification locus).
Looking in McConnell-Robson Noncommutative ...

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votes

**1**answer

245 views

### GCD computation for multiple polynomials and degree of Bezout coefficients

Assuming two polynomials $P_1,P_2 \in \mathbb{Z}_p[r]$ of degree $n$, with no common factors, we know that there exist polynomials $Q_1,Q_2$ s.t.: $Q_1P_1 + Q_2P_2 =1$. From Bezout's identity we also ...

**14**

votes

**1**answer

355 views

### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

**12**

votes

**0**answers

499 views

### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...

**18**

votes

**3**answers

936 views

### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

**18**

votes

**5**answers

612 views

### Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
...

**17**

votes

**3**answers

763 views

### Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?

In general, it seems not known which finite abelian groups are class groups of quadratic number fields.
For imaginary quadratic number fileds, I read that $(\mathbb{Z}/3\mathbb{Z})^3$ is the smallest ...

**19**

votes

**4**answers

1k views

### The sum of same powers of all matrices modulo p

The following is a problem from our department algebra competition for
students:
Non-question.
An experimental-math geek was trying to raise all matrices $17\times17$
over the field with 17 ...

**17**

votes

**4**answers

911 views

### Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?

Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...

**1**

vote

**3**answers

408 views

### From reducible polynomial to an irreducible one

Is there some algebraic construction/extension to make a reducible polynomial over $\mathbb{Q}$ irreducible?
For example: consider the polynomial $x^4-x^3-x^2+1=(x-1)(x^3-x-1)\in \mathbb{Q}[x]$.
Is ...

**2**

votes

**1**answer

138 views

### Diophantic equation from finite semisimple rings

Let $k$ and $k'$ and $n_{1},\ldots,n_{k}$ and $m_{1},\ldots,m_{k'}$ be natural numbers. Let $f_{1}\leq \ldots \leq f_{k}$ and $e_{1} \leq \ldots \leq e_{k'}$ be power primes, such that the following ...

**11**

votes

**1**answer

924 views

### At what point does number theory stop playing with finite rings?

Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for ...

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votes

**2**answers

162 views

### power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m
can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...

**3**

votes

**1**answer

469 views

### When does a polynomial split over Q?

If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?!
In another words, what could you impose on the coefficients to ...

**6**

votes

**1**answer

218 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 ...

**2**

votes

**1**answer

155 views

### equivalence of maximal fields in division algebras

Let D be a division algebra over F, E its maximal field. Is it true that:
1) all such fields are equivalent over F?
2) all such fields are conjugate by inner automorphisms of D?

**8**

votes

**1**answer

428 views

### Diophantine theory of homogeneous cubic polynomials

Arithmetic of quadratic forms over $\mathbb{Z}$ (or lattices theory) has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). ...

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votes

**0**answers

165 views

### Determinant of integer lattice basis of `$L=\{(x_1,\ldots,x_n): a_1x_1+\cdots+a_nx_n=0\}$`

Question: Suppose $\{v_1,\ldots,v_{n-1}\}$ is
an integer basis for the lattice
$$L=\{(x_1,\ldots,x_n)\in\mathbb{Z}^n:
> a_1x_1+\cdots+a_nx_n=0\},$$ where the
$a_i$ are fixed nonzero ...

**3**

votes

**2**answers

337 views

### Effect on Hecke Operator on $\Gamma(N)$ Eisenstein series

Hi.
For Modular Forms for $SL_2(\mathbb{Z})$ there is an easy argument why the (essentially only) Eisenstein series has to be an Eigenform of the Hecke operators. What i am now trying to see is that ...

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votes

**0**answers

243 views

### Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial

Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$.
Let $I=(p_1,p_2)$ be an Ideal of ...

**10**

votes

**5**answers

553 views

### Is complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?

Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h_a=\text{ sum of all monomials of degree ...

**5**

votes

**0**answers

325 views

### Ideal class group isomorphic to $\mathbb{Z}$

Hi everybody,
I wonder if someone could provide me with a simple example of a Dedekind ring whose ideal class group is isomorphic to $\mathbb{Z}$.
The point is I would like the example simple enough ...

**4**

votes

**1**answer

352 views

### integral equivalence classes of quadratic forms

Let $A$ and $B$ two symmetric matrices definite positive over $\mathbb{R}$. Then we say that $A$ and $B$ are integrally equivalent if there exists $Q\in GL_n(\mathbb{Z})$ such that
$A=Q.B.Q^t$ (1)
...

**1**

vote

**1**answer

348 views

### Maximal order in a central simple algebra

Suppose $A$ is a central simple algebra over a field $F$, $\mathcal{O}_F$ is an integrally closed subring of $F$ and suppose $\mathcal{O}_F$ is noetherian. Then an order of $A$ is a $\mathcal{O}_F$ ...

**4**

votes

**4**answers

874 views

### Is there an analogue of finite fields for products of two prime powers?

The collection of prime powers can be characterized in the following way:
There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...

**5**

votes

**2**answers

506 views

### When is $\mathbb{G}_m(R)$ enough to determine $R$?

Say I have a ring, $R$, with 1 which I consider my universe, and I know its group of units $G=\mathbb{G}_m(R)$. Then given a subgroup, $H\le G$, can I determine if there is there a subring $S_H$ such ...

**5**

votes

**3**answers

544 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 ...

**8**

votes

**1**answer

540 views

### Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...

**1**

vote

**1**answer

167 views

### representation of integers as the product of linear forms in three variables

I would like to find all integer triples (x,y,z) such that: $\prod_{\theta}(x + y \theta + z \theta^2)=1$, where $\theta$ runs through the solutions to the cubic $x^3 + x^2 - 2x - 1=0$.
In his book ...

**5**

votes

**0**answers

2k views

### Prime ideals in $\mathbb{Z}[\sqrt{-5}]$ [closed]

First of all, sorry for the noob question, but it's driving me crazy...
I was reading John Stillwell's "Elements of Number Theory" (Springer, ISBN 0-387-95587-9).
In an exercise on page 225, he ...

**5**

votes

**1**answer

484 views

### When is a ring the ring of adeles of some global field

Given a global field $F$, we can construct the ring of adeles. Given a general locally compact ring $R$, when is it isomorphic to the ring of adeles of some global field $F$ and how can I find $F$ in ...

**10**

votes

**2**answers

560 views

### Examples of Galois-invariant central simple algebras which aren't base change?

Suppose $L/K$ is a Galois extension of number fields, with Galois group $G_{L/K}$. Write $\mathrm{Br}(L)^{G_{L/K}}$ for the subgroup of central simple algebras $A/L$ which are Galois-invariant; ...

**12**

votes

**3**answers

957 views

### How to distinguish division algebras from matrix algebras?

Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset ...

**3**

votes

**2**answers

324 views

### Rank of sum of Galois conjugates of a matrix

Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in ...

**1**

vote

**2**answers

477 views

### trace zero elements in algebraic number fields

If we are given an algebraic number field L, and $ \alpha $ is an element of L whose field trace over Q is zero and whose field norm over Q belongs to Z, then does $ \alpha $ necessarily belong to the ...

**28**

votes

**3**answers

2k views

### What is the current status of Agrawal's conjecture?

In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture:
If for coprime integers $n$ and $r$ the equality $(X-1)^n = X^n - 1$ holds in ...

**0**

votes

**0**answers

239 views

### Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + ...

**17**

votes

**4**answers

910 views

### Why are polynomials easier to handle with than integers?

This may seems to be an elementary question, but I found no answers on MO nor google.
I have always heard "polynomials are easier to handle with than integers". For example:
When $n$ is quite ...

**2**

votes

**1**answer

491 views

### Smith Normal Form and lower triangular Toeplitz Matrices

I am working on a undergrad research project with some other guys. Now the conjecture (unrelated to this question) we are trying to prove boils down to a final subproblem:
Let $A (n \times n)$ be a ...

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votes

**4**answers

306 views

### Familiar equations in more general settings

What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...

**3**

votes

**0**answers

183 views

### Localization of power series and module structure

Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable.
Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials.
Let also $\widehat{R}$ be the ring of ...

**5**

votes

**1**answer

468 views

### What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?

What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...

**14**

votes

**3**answers

1k views

### What is interesting/useful about big Witt Vectors?

p-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring A of characteristic p a complete DVR of characteristic 0 with residue ring A generalizing Z_p and F_p.
...