# Tagged Questions

**-3**

votes

**0**answers

46 views

### Why is a principal prime ideal of $\mathrm{PID}[x]$ not maximal? [migrated]

Let $R$ be a PID and let $f(x)\in R[x]$ be an irreducible primitive polynomial. I want to show that the prime ideal $(f)<R[x]$ is not maximal. It would be enough to find a prime $p\in R$ such that ...

**3**

votes

**1**answer

176 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

**1**answer

220 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

**1**answer

240 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

**1**answer

180 views

### $\lambda$-invariant

Please give some hints or references for proving the following -
Let $p$ be an odd prime and $\mathbb{Z}_p$ denotes the $p$-adic integers. Suppose $M$ is a finitely generated torsion ...

**4**

votes

**1**answer

207 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

**1**answer

286 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 ...

**6**

votes

**2**answers

309 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

**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): ...

**0**

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**0**answers

113 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

**1**answer

218 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 ...

**6**

votes

**4**answers

347 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

**3**answers

148 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

**2**answers

288 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

**2**answers

180 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

**1**answer

253 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

**0**answers

77 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**

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**0**answers

135 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

**0**answers

142 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

**1**answer

147 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 ...

**3**

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224 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

**0**answers

48 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**

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**0**answers

94 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.
...

**2**

votes

**1**answer

190 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

**2**answers

446 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 ...

**27**

votes

**3**answers

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

**0**answers

141 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

**0**answers

112 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

**1**answer

410 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

**3**answers

427 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

**2**answers

224 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

**3**answers

410 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

**2**answers

180 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**

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**0**answers

418 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

**1**answer

180 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

**3**answers

357 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

**2**answers

608 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

**0**answers

200 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 ...

**8**

votes

**3**answers

351 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**

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**0**answers

408 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

**1**answer

300 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

**2**answers

450 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

**0**answers

244 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

555 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

**1**answer

248 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

**1**answer

185 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

**1**answer

178 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 ...

**10**

votes

**2**answers

704 views

### About integer polynomials which are sums of squares of rational polynomials…

I have the following question for which I haven't been able to find any reference or proof.
Suppose we know that a univariate polynomial $P(X)$ with integer coefficients is the sum of squares of two ...

**2**

votes

**1**answer

282 views

### Quotient field extension for an incomplete DVR

Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q(R)$, the fraction ...

**2**

votes

**1**answer

232 views

### What does the d-slice of a weighted polynomial algebra look like?

This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves.
...