All Questions
58 questions
2
votes
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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 $A$-...
CommunityBot
- 1
13
votes
2
answers
875
views
Given an irreducible polynomial over $\mathbb{Z}$, how often is it irreducible modulo a prime?
Given a monic irreducible polynomial $f\in\mathbb{Z}[x]$, I'd like to know for how many primes p we have that $f \bmod p$ is irreducible.
In the link: How many primes stay inert in a finite (non-...
- 156k
4
votes
0
answers
116
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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-...
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3
votes
1
answer
206
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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 ...
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74
votes
3
answers
7k
views
Is there a "purely algebraic" proof of the finiteness of the class number?
The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching ...
- 50.6k
6
votes
0
answers
629
views
On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"
I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...
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1
vote
1
answer
91
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Duogenic quartic rings
Recall that a commutative, unital ring $R$ of finite rank which is isomorphic to $\mathbb{Z}^n$ for some $n \geq 1$ as $\mathbb{Z}$-module is said to be monogenic if there exists an element $\gamma \...
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1
vote
0
answers
77
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Subrings of rings of integers of quartic fields having prime index and a specific property
Let $L$ be a $S_4$ or $A_4$ quartic field, and $\mathcal{O}_L$ its ring of integers. Let $K$ be its cubic resolvent field, which necessarily has the same discriminant as $L$. If $\mathcal{O}_L$ has a ...
- 26.9k
6
votes
2
answers
329
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Algebraic numbers which prescribed degree which does not belong to some fields
In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
- 20.4k
6
votes
1
answer
894
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Brauer group of rational numbers
Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
- 50.6k
4
votes
0
answers
82
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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{...
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1
vote
1
answer
172
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Completion reducing to localization on Noetherian rings
It is quite easy to show that if $A$ is a Dedekind domain and $\mathfrak{p}\in \operatorname{Spec} A$, then if $A_{\mathfrak{p}}$ is the completion of $A$ at $\mathfrak{p}$ and $A_{(\mathfrak{p})}=(A\...
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5
votes
1
answer
335
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About the structure of unit groups appearing in number theory
I think the following statement is not true in the general situations, but consider it:
$R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...
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2
votes
0
answers
311
views
Degree $8$ cyclic extension over $\mathbb{Q}$
Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
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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 ...
2
votes
1
answer
193
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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 $\Lambda_d$-...
10
votes
3
answers
1k
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What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
- 63.9k
3
votes
1
answer
149
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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 ...
- 37k
4
votes
0
answers
402
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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
1
vote
0
answers
107
views
Topologically finitely generated non-abelian isomorphic absolute Galois groups
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 topologically finitely generated, non-abelian and ...
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3
votes
1
answer
184
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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 $L$ ...
- 2,051
9
votes
0
answers
441
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Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)
I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
- 639
5
votes
1
answer
293
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The Kronecker--Hurwitz property for rings of integers in global function fields
In Ireland and Rosen's book on number theory they give a proof of the finiteness of the class group of a number field which they attribute to Hurwitz, but which is essentially due to Kronecker (as I ...
- 3,813
2
votes
1
answer
262
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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 $...
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2
votes
0
answers
728
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On Serre's "Local fields"
While I was reading J.-P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to ...
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2
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0
answers
491
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Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
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9
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2
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815
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Ideal norm in orders
Let $\overline{T}$ be a Dedekind ring such that $\overline{T}/\overline{I}$ is finite for every nonzero ideal $\overline{I}$ of $\overline{T}$. Let $T$ be a subring of $\overline{T}$ with the same ...
- 7,893
6
votes
0
answers
119
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Norm forms, slicing, and ideal classes
Let $K$ be a number field, which we may suppose satisfies $n = [K : \mathbb{Q}] \geq 3$. Let $\mathcal{O}_K$ be the ring of integers of $K$, and let $\{\omega_1, \cdots, \omega_{n}\}$ be a basis of $\...
- 26.9k
2
votes
1
answer
216
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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{...
- 105k
2
votes
0
answers
300
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Elliptic curves: about a passage in J. Silverman's "Advanced topics of elliptic curves"
Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage ...
- 105k
0
votes
1
answer
495
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Roots of unity, in the completion of the ring of integers of a number field, at a prime ideal
Let $p \in \mathbb{Z}$ be a prime and consider the number field $k = \mathbb{Q}[x]/(x^{(p^2 -1)/2} - p)$. We shall denote by $O_k$ the ring of integers of $k$. Let $\beta \in O_k$ be such that $\beta^{...
- 50.6k
8
votes
2
answers
2k
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Fermat's Last Theorem in finite fields
Consider the finite field $\mathbb{F}_q$. Schur (1916) proved that, given $n$, when the field is sufficient large, this equation,
$$x^n+y^n= z^n$$
always has a nontrivial solution.
What conditions ...
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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,...
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1
vote
0
answers
295
views
Proper ideals are invertible
I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $...
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2
votes
0
answers
93
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The prime spectrume of integral-valued polynomial ring
Let $ D $ be an integral domain with quotiont field $K $ and let $Int (D) $be the set of all integral-valued polynomials on $D $, that is, $ Int (D):=\{f \in K[x]\mid f (D) \subseteq D\} $. The ...
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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
1
vote
1
answer
934
views
Bound on number of proper ideals of norm equal to n
I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand:
Let $d$ be a positive non-square interger and set let $K = \...
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8
votes
3
answers
1k
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Sufficient conditions for a polynomial to be reducible over the integers
There are several well-known criteria for a polynomial with integer coefficients to be irreducible over $\mathbb{Z}$, e.g., Eisenstein's criterion. I'm looking for the opposite: other than ...
4
votes
0
answers
144
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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= \...
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12
votes
1
answer
656
views
What kind of arithmetic information does the ring of integers in an infinite extension carry?
The fact that the ring of integers in a finite extension of $\Bbb Q$ is a Dedekind domain and purely algebraic properties of Dedekind domains are absolutely essential for algebraic number theory. So ...
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26
votes
1
answer
4k
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Underlying structure behind the infamous IMO 1988 Problem 6
This is the infamous Problem 6 from the 1988 IMO which has recently been popularised by the YouTube channel Numberphile:
Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{...
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1
vote
0
answers
133
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Finding Generators of an Ideal Over Number Field? [closed]
Is there any way or algorithm to find generators of an ideal over number field? (A algorithm that can be implemented and not expensive)
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24
votes
2
answers
2k
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Can one prove the elementary divisor theorem for PIDs by elementary matrix operations?
The elementary divisor theorem was originally proved by a calculation on integer matrices, using elementary (invertible) row and column operations to put the matrix into Smith normal form. That is ...
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14
votes
1
answer
695
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$\mathbb{Z}$-module structure of the subring generated by an algebraic number
Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
- 148k
1
vote
0
answers
164
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Combinatorial splitting in number rings
The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring.
Take an arbitrary non empty ...
CommunityBot
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2
votes
1
answer
216
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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 $SL_n$ ...
CommunityBot
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8
votes
0
answers
446
views
Capitulation of ideal classes in general Dedekind Domains
I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up:
Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify ...
CommunityBot
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4
votes
2
answers
2k
views
Prime ideals in the ring of algebraic integers
Let $m(x) = x^n + a_{n-1}x^{n-1} + \dots + a_1 x+ a_0$, $a_i \in \mathbb{Z}$, be an irreducible polynomial over $\mathbb{Q}$ and $K = \mathbb{Q}[x] / {m(x)\mathbb{Q}[x]}$, so $K$ is an algebraic ...
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13
votes
2
answers
1k
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Number of polynomials whose Galois group is a subgroup of the alternating group
Let $f = x^n + a_{n-1}x^n + \cdots + a_0$ be a monic polynomial of degree $n \geq 2$ with integer coefficients. By $\text{Gal}(f)$ we mean the Galois group over $\mathbb{Q}$ of the Galois closure of $...
- 96.4k
1
vote
0
answers
189
views
What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]
Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...
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