3
votes
1answer
149 views

Linear dependency of real numbers with integer coefficients adding up to zero [closed]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both ...
1
vote
1answer
116 views

General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset. Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...
3
votes
0answers
99 views

Finding an integral basis for the lattice of the form $\mathbb{Z}^J \cap \mathbf{p}^{\perp}$

Let $\mathbf{p}$ be a primitive point in the lattice $\mathbb{Z}^J$ and denote the $J-1$ dimensional vector space $V = \mathbf{p}^{\perp} \subseteq \mathbb{R}^J$. Let $\Lambda' = \mathbb{Z}^J \cap V ...
0
votes
0answers
50 views

Extensions on Higher-dimensional local fields

I have the following question: Let $M/L$ b a finite extension of n-dimensional local fields and $t_1,\dots, t_n$ a system of local parameters of $L$ with valuation $v$. Let us fix an $1\leq i \leq ...
1
vote
0answers
37 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 ...
0
votes
0answers
64 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
0answers
110 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
0answers
131 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 ...
5
votes
1answer
214 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
0answers
84 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
2answers
313 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 ...
2
votes
0answers
103 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 ...
0
votes
1answer
266 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
1answer
376 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
0answers
512 views

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

How to explain the patterns in the matrix defined in Fran├žois Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
20
votes
4answers
1k 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
5answers
651 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: ...
18
votes
3answers
808 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
4answers
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 ...
19
votes
4answers
943 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
3answers
430 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
1answer
141 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
1answer
938 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 ...
0
votes
2answers
163 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
1answer
479 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
1answer
222 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
1answer
161 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
1answer
434 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). ...
1
vote
0answers
192 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 ...
4
votes
2answers
390 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 ...
0
votes
0answers
249 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
5answers
580 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
0answers
338 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 ...
5
votes
2answers
475 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
1answer
365 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
4answers
897 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
2answers
518 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
3answers
562 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
1answer
544 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
1answer
168 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
0answers
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
1answer
488 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
2answers
571 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
3answers
977 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
2answers
328 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
2answers
507 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
3answers
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
0answers
252 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
4answers
913 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
1answer
500 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 ...