-1
votes
1answer
123 views

When is a local subring of a number field a valuation ring?

Do we have some good examples of local subrings of number fields which are not valuation rings? Do we have an easy criterion for determining whether a local subring of a number field is a valuation ...
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 ...
4
votes
1answer
439 views

A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
4
votes
1answer
209 views

Circulant matrix with integer entries and determinant 1 or -1

CONJECTURE Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$th column of $A$ is obtained by applying the permutation ...
3
votes
1answer
233 views

operations on ideals in a subring of number field

For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$, does this equality hold in general? $(I+J) \cap K = (I \cap K) + (J \cap K)$ I have no counterexample yet but I couldn't prove ...
4
votes
1answer
254 views

Torsors and the fpqc topology

Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want ...
3
votes
1answer
262 views

Splitting as $\mathbb{F}_p[[X]]$-modules

Let $A$ be a finitely generated torsion $\mathbb{Z}_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB ...
1
vote
1answer
333 views

Iwasawa algebra [closed]

Let $\mathbb{Z}_p$ denotes the $p$-adic integers for a prime $p$. Suppose $M$ is a finitely generated torsion $\mathbb{Z}_p[[T]]$-module such that $\mu(M)=0$. Then $M/pM$ and $M[p]$($p$-torsion ...
5
votes
1answer
213 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 ...
3
votes
1answer
293 views

Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...
7
votes
2answers
321 views

convergence in Z-hat; modulo prime power

The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16). Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$ by $a_0=b, a_{n+1}=2^{a_n}$. Prove that ...
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): ...
0
votes
0answers
117 views

Action on C[[X,Y]]/f(X,Y) giving complete intersection quotients

Let $R \colon\!= {\Bbb C}[[X,Y]]/(f(X,Y))$ be a complete local ring of Krull-dimension $1$. Assume that we have an action of $\Bbb Z$ on $R$ such that fixed elements by $\Bbb Z$ in $R$ are only ...
0
votes
1answer
231 views

Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other. Let ${\Bbb Q}_{\infty}$ be the unique ...
7
votes
4answers
357 views

On the fixed point of automorphism of F_3[[T]]

Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq ...
0
votes
3answers
149 views

Behavior of duality under pull-back

I have a technical question on commutative algebra. I am not an expert in the subject, and I would like to know if there are "typical conditions" making the following possible. Let $\varphi:R\to S$ ...
0
votes
2answers
295 views

Pontryagin dual

Suppose $M$ is a $Z_p[[T]]$-module and $\widehat{M}$(the Pontryagin dual of $M$) is a finitely generated torsion $Z_p[[T]]$-module. How to prove that $\widehat{M}$ has $\mu$-invariant zero ...
0
votes
2answers
186 views

0-dimensional Gorenstein local ring.

Assume the following condition for the ring T = F_p[[X,S]]/I: Condition 1. T is NOT a zero ring. Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements. Then, is T a ...
0
votes
1answer
266 views

Iwasawa invariants

Suppose $M$ is a finitely generated torsion $Z_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are ...
6
votes
0answers
78 views

Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
1
vote
0answers
141 views

Ergod Theorem for $\mathbb{F}_{3}[[X,S]]$

Assume we have the automorphism on $2$-variable power series ring $\mathbb{F}_{3}[[X,S]]$ over finite field $\mathbb{F}_3$ as follows: $σ: S \longrightarrow S + S^3$ $σ : X \longrightarrow X + S + ...
0
votes
0answers
146 views

A question on binary polynomials

This is probably a well-known result but I was not able to find a reference on my search. My question concerns general polynomials $f(x,y) \in \mathbb{Z}[x,y]$ such that $f$ cannot be written as a ...
1
vote
1answer
150 views

Polynomial analogue of “prime independence”

In number theory a well-known fact is that congruence modulo distinct primes are 'independent'. That is, to know that $n \equiv a \pmod{p}$ does not change the probability as to what $n \equiv x ...
2
votes
0answers
236 views

PAC field : Algebraically closed field :: ? : Henselian local ring

I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity. I'd want to call a DVR $(R,\mathfrak{m})$ ...
2
votes
0answers
54 views

different and discriminant for finite invariants

Let $k$ be an algebraically closed field. Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...
2
votes
0answers
97 views

Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$

The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked. ...
3
votes
1answer
195 views

Duality for rank one modules over a number ring

Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that ...
0
votes
2answers
451 views

An extremal combinatorics problem over Finite Rings

Cross Posting from: http://math.stackexchange.com/questions/462016/a-combinatorics-problem-over-finite-rings Consider the set $S$ of all non-zero vectors over $\Bbb Z_{q}$ of length $r$ whose ...
28
votes
3answers
1k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
3
votes
0answers
143 views

Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb ...
2
votes
0answers
117 views

Lattices as invertible modules.

I have asked this question in Math Stack exchange but got no answer. Maybe it fits Mathoverflow better. All rings below are assumed to be Noetherian. Let $E$ be an etale algebra over ...
11
votes
1answer
416 views

First order decidability of rings vs Diophantine decidability

Are there known (preferably ``concrete'') examples of a ring $R$ (commutative, with 1) such that: $\bullet$ the first order theory of $R$ is undecidable, but $\bullet$ the positive existential (= ...
1
vote
3answers
431 views

Can you compute the quotient set below?

Let $K$ be a field of characteristic $2$ ($2$ is very important in the statement -- otherwise I can do it myself :) ). On the set $K \times K$ we define the following equivalent relation: $(a, b) ...
8
votes
2answers
229 views

A criterion for freeness over a local ring

Let $A=K[[X_1,\dots,X_n]]$ where $K$ is a field. Let $M$ be a finitely generated torsion-free $A$-module, such that for all $k$, the $A[1/X_k]$-module $M[1/X_k]$ is free of rank $d$; for every $i ...
1
vote
3answers
429 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 ...
1
vote
2answers
190 views

Transformation of a bivariate polynomial into a homogeneous one

For a given a bivariate polynomial $P(x,y)$ with rational coefficients: Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...
12
votes
0answers
425 views

How many ways can one cover $\mathbb Q_p$ with the images of polynomials?

Define a finite set of polynomials over a field $K$ to cover $K$ if the images of the polynomials, viewed as functions from $K$ to itself, have union the whole set. Define a minimal cover to be a ...
4
votes
1answer
181 views

Is there a local-global principle for integral Laurent series ?

Motivation: A real number is rational iff its decimal expansion is periodic (by "periodic" I mean periodic after some steps). Similar, a p-adic number is rational iff its p-adic expansion is periodic. ...
4
votes
3answers
392 views

Orders of Number Fields

Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order. $\textbf{Questions:}$ $\newcommand{\Spec}{\textrm{Spec }}$ $\newcommand{\cO}{\mathcal{O}}$ ...
6
votes
2answers
662 views

henselization and completion

This might not be a question appropriate for this forum, I apologize in this case... Is it true that any element of the completion of a valued ring $R$ that is algebraic over the field of fractions of ...
3
votes
0answers
206 views

Question about witt vector of some ring

Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And ...
9
votes
3answers
368 views

Proving finite generation by tensoring with $\mathbb{R}$

In Chapter III, Theorem 7.4 of The Arithmetic of Elliptic Curves (first edition), Silverman gives the following lemma and proof: Lemma: Let $M \subset Hom(E_1, E_2)$ be a finitely generated ...
13
votes
0answers
429 views

Bloch-Kato conjecture and Wiles' numerical criterion

I already asked this question some days ago on http://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response. In the introduction ...
4
votes
1answer
306 views

What happens to factors of the resultant upon specialization?

Let $f, g$ be two polynomials in $S[t]$ where the coefficient ring is $S = \mathbb{C}[a_1..a_n]$. The resultant of $R(f,g)$ gives some measure as to whether or not $f$ and $g$ share a common factor. ...
1
vote
2answers
467 views

Is an elementary symmetric polynomial an irreducible element in the polynomial ring?

Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $e_a$ denotes the elementary symmetric polynomials of degree $a$ in $S$. For $n=2$: $e_1=x_1+x_2$; $e_2=x_1x_2$. For $n=3$: ...
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 ...
2
votes
1answer
255 views

Is there a Dirichlet Unitary Unit Theorem?

Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available. Assume the order has an involution. For example, ...
2
votes
1answer
188 views

Extensions of truncated Witt vectors

Let $k$ be a perfect field of characteristic $p>0$. For any positive integers $n$, let $W_n(k)$ be the truncated Witt vectors of length $n$ with coefficients in $k$. For any positive integers ...
4
votes
1answer
180 views

L-series/density theorems for Dedekind domains

If $\mathcal{O}$ is the ring of integers of a number field, then the Hecke-L-series for a character $\chi$ of the class group is defined as $$L(\chi,s) = \sum_{\mathfrak{a} \neq ...