All Questions
Tagged with nt.number-theory ac.commutative-algebra
360 questions
4
votes
2
answers
405
views
Transitivity of discriminant for flat algebras
Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...
- 43
4
votes
1
answer
457
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 $(1,2,..,n)...
- 86
4
votes
1
answer
334
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
4
votes
1
answer
798
views
Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?
Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$.
Let $L \subset \hat{K}$ be a separable finite ...
- 41
4
votes
1
answer
367
views
Do n-th Witt polynomials generate {P | P' is divisible by n} ?
EDIT: Proved it on my own. It easily follows from the Witt integrality theorem. Sorry for posting.
Let $P\in\mathbb{Z}\left[\Xi\right]$ be a polynomial (where $\Xi$ is a family of symbols that we use ...
- 33.8k
4
votes
1
answer
393
views
ring structure on free abelian groups
Let $\mathbb{Z}^{n}$ be the free abelian group of rank $n$.
A ring structure on $\mathbb{Z}^{n}$ is a choice of a unit element $e\in \mathbb{Z}^{n} $ and a bilinear map $m:\mathbb{Z}^{n}\otimes_{\...
- 1,613
4
votes
1
answer
261
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 0}\frac{\chi(\...
- 16.2k
4
votes
1
answer
728
views
matrix congruence and smith normal form
Fixed $n \geq 2$ and consider $A,B \in GL(n,\mathbb{Z}).$ We know that we have the Smith normal form. One can find $U, V \in SL(n,\mathbb{Z})$ such that $A=UDV.$ So as $B$. The Smith normal form is ...
- 145
4
votes
2
answers
490
views
Krull dimension of completions in non-noetherian setting (especially completed perfections)
What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology?
An example of the sort of "nice" topological ring I'm looking for is a ...
- 3,058
4
votes
1
answer
240
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 $R=\textrm{End}...
- 2,406
4
votes
1
answer
467
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 ...
- 10.9k
4
votes
2
answers
493
views
Explicit Bézout cofactors
$\DeclareMathOperator\lcm{lcm}$This is a rather severe revision of a question I asked recently. We know over the integers that $\gcd(a^2,b^2)=\gcd(a,b)^2$. We might prove this via unique factorization....
- 30.1k
4
votes
0
answers
116
views
The criterion for dimensional conjecture for universal Galois deformation rings
I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
- 863
4
votes
0
answers
171
views
Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$
This is related to an open problem.
Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.
Let $S$ be the set of degree 2 nilpotent elements ...
- 25.4k
4
votes
0
answers
389
views
Intersection of principal ideals is not principal
This may be an easy question for the right people, but I did not find an answer anywhere. I am trying to figure out what one can say about the ideal $a\mathbb{Z}[\lambda] \cap b \mathbb{Z}[\lambda]$ ...
- 135
4
votes
0
answers
82
views
The index of an order defined by a binary form
In his well-known paper, Nakagawa generalized a construction due to Birch and Merriman to arbitrary binary forms and orders. In particular, his construction gives a canonical algebraic order $\mathcal{...
- 26.9k
4
votes
0
answers
244
views
Torsionness of the kernel of the pullback map of Picard groups of a normalization map
Let $X$ be a (irreducible) projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of ...
- 151
4
votes
0
answers
211
views
Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
4
votes
0
answers
236
views
Is this property of polynomials generic?
Let $n \geq 2$, and consider a polynomial $f$ in $n$ variables, say over a field $K$ of characteristic 0. Recall that $f$ is geometrically irreducible if $f$ is irreducible over the algebraic closure ...
- 26.9k
4
votes
0
answers
402
views
Is every Dedekind domain the integral closure of some principal ideal domain?
I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ ...
- 1,053
4
votes
0
answers
144
views
Does a countably generated $\mathbb{Q}$-algebra inject into some $p$-adic field?
Let $K$ be a subfield of $\mathbb{C}$. If $K$ is finitely generated over $\mathbb{Q}$, then $K$ injects into $\mathbb{Q}_p$ for some $p$.
Assume that $K$ is countably generated, i.e., $K= \...
- 71
4
votes
0
answers
166
views
Frobenius number of a local ring
Let $(R, \mathfrak{m})$ be a (complete) Noetherian local ring (domain) of positive dimension. For each $\mathfrak{m}$-primary ideal $I$, let $e(I)$ be the multiplicity of $I$. Let
$$\mathcal{A}(R) = \...
- 2,773
4
votes
0
answers
191
views
Chevalley-Warning for finite rings: the degree of a non-polynomial
$\def\F{\mathbb F}$
$\def\Z{\mathbb Z}$
One reason that Chevalley-Warning theorem is that amazingly useful is the fact that for a finite field $\F$, any function from $\F^n$ to $\F$ is a polynomial. ...
- 23k
4
votes
0
answers
152
views
On the computational complexity of the Hilbert polynomial of numerical semigroup rings
Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that $\ell(R/\...
- 2,773
4
votes
0
answers
345
views
Domains with prime ideal theorems
Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm Spec}(D):N({\...
- 441
4
votes
0
answers
338
views
What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
- 20.5k
3
votes
7
answers
4k
views
How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
- 179
3
votes
2
answers
810
views
What is the divisibility theory for Bezout Domains?
There are many facts about integer gcds which can be proved by appealing to unique prime factorization (up to sign). for example $\gcd(a^2,b^2)=\gcd(a,b)^2$. One way to get the machinery (if that is ...
- 30.1k
3
votes
1
answer
1k
views
Does totally ramified extension really exist?
The answer is certainly "Yes", but this is the problem I met in Algebraic Number Theory by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are ...
- 95
3
votes
2
answers
382
views
Localization at multivariate monic polynomials
Let $R$ be a ring and consider a monomial order $<$ on $R[X_1,\ldots,X_n]$. A nonzero polynomial $f \in R[X_1,\ldots,X_n]$ is said to be monic if its leading coefficient with respect to $<$ is $...
3
votes
1
answer
195
views
Solutions to nonhomogeneous quadratic equation mod $N$
Is there any way to find non-trivial solutions to the equation $x^2 + y^2 - x \equiv 0 \mod{N}$? There are clearly several trivial solutions, for example $(x, y) = (0, 0), (1, 0), (2^{-1}, 2^{-1}), (2^...
- 1,703
3
votes
1
answer
184
views
Non-abelian isomorphic absolute Galois groups of fields of different characteristic
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ ...
- 55
3
votes
2
answers
338
views
Isomorphism between finite algebras over ${\Bbb Z}_p$
Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...
- 563
3
votes
1
answer
400
views
Invertibility of a matrix whose entries are certain binomial coefficients
Let $l$ be a positive integer. Does the matrix
$$
M_l \ := \ \left( \binom{l-(2p+1)}{j} \right)_{0\leq p,j \leq[(l-1)/2]}
$$
have nonzero determinant?
- 63
3
votes
2
answers
201
views
Linear polynomials in units of number fields
I would be thankful for a reference to any result that says "how often" an equation of the form $$c_1x_1 + c_2x_2 + ... + c_nx_n = 0,$$
where $n$ is fixed, $c_1, ..., c_n \in \mathcal{O}_K$ are ...
- 704
3
votes
1
answer
470
views
Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?
The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian ...
- 193
3
votes
1
answer
947
views
Module of Kahler differentials of rings of integers of number fields
How does one prove that if $L/K$ is an extension of number fields with rings of integers $B/A$, then the module of Kahler differentials $\Omega^1_{B/A}$ can be generated by one element as a $B$-module?...
- 5,019
3
votes
1
answer
136
views
Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals
Suppose $K=\mathbb{Q}(X_1,\dots,X_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be ...
- 19.6k
3
votes
1
answer
336
views
If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?
Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...
- 671
3
votes
1
answer
420
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 F}...
- 87
3
votes
1
answer
475
views
Can the factorisation of (p) in a number field K be described by the minimal polynomial of a primitive element?
Let $K$ be a number field, with ring of integers $O_K$, and let $\alpha\in O_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, ...
- 2,406
3
votes
1
answer
390
views
The kernel from $A_\mathrm{inf}$ to $\mathcal{O}_{\mathbb{C}_K}$
I tried to understand this paper on page 31.
Let $K$ be an finite extension of $\mathbb Q_p$ and $\overline{K}$ be its algebraic closure; $\mathcal{O}_{\overline{K}}$ is the ring of integers of $\...
- 275
3
votes
2
answers
549
views
Primary structures in $\mathbb Q$
I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
- 7,500
3
votes
1
answer
341
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 \...
- 193
3
votes
1
answer
347
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 $a,b$,...
- 33
3
votes
1
answer
206
views
Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\ad{ad}\DeclareMathOperator\gen{gen}$Let $p$ be a prime and $G$ be a profinite group such that the pro-$p$ completion of every open subgroup is ...
- 863
3
votes
1
answer
149
views
How to compute the intersection of an ideal with the maximal order of a subfield?
I asked this earlier on math.stackexchange but I think this is a better place for this question.
Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be ...
user106850
3
votes
1
answer
320
views
Decision problem about the existence of solution for an integer matrix equation
Given $A,B,C \ $ integer matrices of dimensions $l \times m$, $l \times n$ and $l \times m$, we want to decide (algorithmically) about the existence of $X$ (unimodular) and $Y \ $ integer matrices ...
- 61
3
votes
1
answer
200
views
Kernel of a map of Tate algebras
Let $A$ and $B$ be a pair of noetherian Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow ...
- 2,907
3
votes
1
answer
776
views
On the coherence of a Néron-ring
Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
- 563