All Questions
Tagged with nt.number-theory ac.commutative-algebra
360 questions
3
votes
1
answer
307
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 ...
- 367
3
votes
0
answers
375
views
On the analogy between $p$-derivations and derivations
$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such ...
- 63.9k
3
votes
0
answers
280
views
The closed unit adic disk
I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...
- 2,907
3
votes
0
answers
182
views
Is there any analogue of $\pi$ for non-Archimedean Euclidean fields?
Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the ``Euclidean'' plane $F^2$ endowed with the ...
- 41.8k
3
votes
0
answers
139
views
Dirichlet unit theorem for finite rings
Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements ...
3
votes
0
answers
69
views
Division algorithm for multivariable power series
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Consider the ring $R=\mathbb{Z}_p[[T]]$. Let $f,g \in R$ and assume that $f=a_0+a_1T+...$ with $a_i \in p\mathbb{Z}_p$ for $0\le i \le n-1$, but $...
- 429
3
votes
0
answers
110
views
Efficient computation of "higher order" Jacobi symbols
Let $N> 1$, and suppose $a \in (\mathbb{Z}/N\mathbb{Z})^{\times}$. There are efficient algorithms to compute the Jacobi symbol $\left(\frac{a}{N}\right)$ using quadratic reciprocity. Fix $k \geq 1$....
- 1,703
3
votes
0
answers
311
views
Eichler orders in a certain quaternion algebra
Let us consider a totally real number field $K$ such that $[K \colon {\Bbb Q}] = {\mathrm{odd}}$. We shall consider the quaternion algebra $D$ over $K$ such that $D$ splits everywhere at finite places ...
- 781
3
votes
0
answers
78
views
Polynomial equations parametrized by binary forms
Consider the equation
$$\displaystyle Ax^p + By^q = Cz^r, A,B,C \in \mathbb{Z}, \gcd(x,y,z) = 1, p,q,r \geq 2.$$
When $p^{-1} + q^{-1} + r^{-1} > 1$, the above equation is called spherical and ...
- 26.9k
3
votes
0
answers
222
views
Mod p reduction of geometrically irreducible polynomials
Let $f\in \mathbb Z[t,x]$ be a polynomial of positive degree that is irreducible over $\overline{\mathbb Q}[t,x]$. Is it true that for all but finitely many primes $p$ the reduced polynomial $f_p\in \...
- 101
3
votes
0
answers
309
views
Witt vectors with $p$-torsion
Let $R$ be a commutative ring, $W(R)$ is a ring of Witt vectors over $R$. Can you give an example of a ring $R$ such that $W(R)$ has $p$-torsion?
- 2,305
3
votes
0
answers
297
views
Flatness of R[X]/I over R
In the famous paper Simple flat extensions, Journal of Algebra Volume 16, Issue 1, September 1970, p. 105-107, Wolmer V. Vasconcelos proves the following
Theorem (Vasconcelos). For a noetherian ...
- 781
3
votes
0
answers
213
views
Galois action on local deformation ring
Let ${\Bbb Q}_p$ be a local field. For a prime $q \not= p$, we consider an irreducible residual Galois representation
$\overline{\rho} \colon {\mathrm{Gal}}(\overline{{\Bbb Q}}_p/{{\Bbb Q}_p}) \to \...
- 781
3
votes
0
answers
95
views
Sign of bivariate polynomial evaluated over two algebraic numbers
I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
- 151
3
votes
0
answers
391
views
Are prime ideals of finite height in the powers series ring in infinitely variables finitely generated?
Let $A:= {\mathbb F}_p[[X_1,...,X_∞]]$ be the infinitely many variables formal power series ring over ${\mathbb F}_p$, which is UFD.
Consider an arbitrary prime ideal $P$ of $A$ such that the height ...
- 781
3
votes
0
answers
181
views
Factorization of linear recurrences
For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that
$$a(n+k) = r_1 a(n+...
user40023
3
votes
0
answers
175
views
polynomial relations between modular functions
$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$
We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
- 10.7k
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
3
votes
0
answers
314
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 $R_{\infty}=...
- 709
2
votes
1
answer
733
views
Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?
Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$.
Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\...
user76479
2
votes
3
answers
2k
views
Algebraic extensions of p-adic closed fields
I have been working with p-adically closed fields and there are two results that are used time and times again in what I am reading, but I cannot find any references where they are proved...
The ...
2
votes
1
answer
470
views
Polynomials with no multiple root
Let $a,d$ be polynomials of $\mathbb Z[X]$ with $\deg a>\deg d\ge0$ and $P$ be a polynomial of $\mathbb Z[X]$. Consider an infinite sequence of integers $(\lambda_n)_n$. Can one assert there exists ...
- 3,998
2
votes
1
answer
223
views
Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring
Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
- 423
2
votes
2
answers
399
views
What fraction of polynomials with integer coefficients are indecomposable?
It is well-known that "most" integers are composite: the Prime Number Theorem tells us that only about $1/\log(N)$ of the integers in the interval $1 \ldots N$ are prime. For polynomials, ...
- 1,703
2
votes
1
answer
323
views
Coprime multivariate polynomials
Let ${\bf R}$ be a gcd domain, $n \geq 2$, $k \in \mathbb{N}^*$, and $f,g \in
{\bf R}[X_1,\ldots,X_n]$. Supposing that $f$ and $g$ are coprime in ${\bf R}[X_1,\ldots,X_n]$, that is, $\gcd(f,g)=1$, ...
2
votes
1
answer
429
views
0 dimensional Dedekind domain?
It seems that the ratio of those authors allowing a field to be a Dedekind domain to those who do not is almost 50 - 50. Why such a bewildering lack of consensus for such an elementary notion?
- 2,857
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 23
2
votes
3
answers
816
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 ...
- 151
2
votes
1
answer
381
views
Reduced complete Tate ring which is not uniform?
Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
- 217
2
votes
1
answer
745
views
Motivation to study the order theory (ring theory)
I'm currently reading a paper of Georges Gras on the Reflection Principle.
The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting,...
- 1,013
2
votes
1
answer
423
views
Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?
In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...
- 173
2
votes
1
answer
458
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 ...
- 7,609
2
votes
1
answer
365
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, ...
- 151
2
votes
1
answer
131
views
Adjacent reducible polynomials
Let $P[X_1, X_2, \ldots, X_N]$ be a reducible polynomial in $\mathbb{Z}[X_1, X_2, \ldots, X_N]$ such that $P[X_1, X_2, \ldots, X_N] + 1$ is also reducible. What (if anything) can we say about $P$?
One ...
- 1,703
2
votes
1
answer
262
views
How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?
I am describing the question details, though the main question is short as below.
Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $...
- 930
2
votes
1
answer
251
views
Étale fibration for $K[[X_1,...,X_n]]$
Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
- 563
2
votes
1
answer
160
views
What is the cokernel of $O_S \to F_\infty/O_\infty$?
Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ ...
user81900
2
votes
2
answers
291
views
Decomposition and valuation rings
I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
- 145
2
votes
1
answer
159
views
Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
- 253
2
votes
1
answer
158
views
Computing the minimal polynomial of roots of polynomials with algebraic coefficients
Let $p(x) = \sum_{i=0}^{n} c_i x^i$ with $c_i \in \mathbb{A}$ with $q_i(c_i) = 0$ and all $q_i \in \mathbb{Q}[x]$ being minimal polynomials of the coefficients.
Let $r$ be a zero of $p(x)$. Is there ...
- 21
2
votes
1
answer
193
views
Existence of non-zero pseudo-null submodules
Let $p$ be a rational prime, and let $\Lambda_d$ be the Iwasawa algebra in $d$ variables, i.e. $\Lambda_d = \mathbb{Z}_p[[T_1, \ldots, T_d]]$. Let $A$ be a finitely generated and torsion $\Lambda_d$-...
- 81
2
votes
1
answer
216
views
A problem about an unramified prime in a Galois extension
I asked this question in MathStackExchange, but I didn't receive any answer.
Let $K/\mathbb{Q}$ be a Galois extension of degree $n$, and denote its ring of integers by $\mathcal{O}_K$. Let $\mathfrak{...
2
votes
1
answer
161
views
How to classify solutions to the following equations?
I would like to classify the sets of integers $a_{1},...,a_{n}$ that satisfy the following two equations.
$$\sum_{k=1}^{n}a_{k}\equiv 0\mod 2$$
$$\sum_{i\neq j}a_{i}a_{j}=0$$
For example, if $n=3$, I ...
- 1,150
2
votes
1
answer
329
views
Heights of contracted ideals
Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...
- 563
2
votes
1
answer
216
views
Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme
Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ ...
- 7,609
2
votes
1
answer
399
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
255
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.
...
- 33.8k
2
votes
0
answers
182
views
Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
- 29
2
votes
0
answers
112
views
Invariant factors and commuting matrices over a discrete valuation ring
$\DeclareMathOperator\Im{Im}\DeclareMathOperator\Ker{Ker}$Let $A$ be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T$, $Y^T$ be ...
2
votes
0
answers
118
views
polynomials with no repeated factors
Assume that $F(x_1,\ldots, x_n)$ is a polynomial with integer coefficients that is "square-free" over $\mathbb Q$, i.e. it does not have repeated polynomial factors whose coefficients are in ...
- 3,062