All Questions
32 questions
0
votes
1
answer
223
views
On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
5
votes
1
answer
335
views
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 ...
5
votes
0
answers
93
views
Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules
Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
- 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
12
votes
2
answers
2k
views
Subsets of the integers which are closed under multiplication
Let $S$ be a subset of the integers which is closed under multiplication. There are many possible choices of $S$:
$S = \{-1, 1\}$.
$S$ is the set of integers of the form $a^k$, where $a$ is fixed and ...
- 1,703
6
votes
1
answer
328
views
Algebra with a certain abelian group as the multiplicative group
Let $A$ be an abelian group. Are there an algebra $\mathfrak{X}(A)$ s.t. the multiplication group is isomorphic to A ? i.e.
$$
\mathfrak{X}(A)^{\times} \simeq A.
$$
For example, for $A=\mathbb{Z}/4\...
- 479
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
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
1
vote
0
answers
310
views
Primes of the power series rings
Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection
\begin{equation*}
\psi_{n,n-1} \colon A_n \...
- 563
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
5
votes
0
answers
185
views
$p$-adic valuation in the ring $\mathbb{Z}/p^k\mathbb{Z}$
Assume $p$ is a prime number, $M$ be a non-negative integer and denote by $(\mathbb{Z}/p^M\mathbb{Z})^*$ the units of $\mathbb{Z}/p^M\mathbb{Z}$. Now consider the partition of $\mathbb{Z}/p^M\mathbb{Z}...
- 3,062
10
votes
0
answers
854
views
Scholze's infinite to finite type ring theory reductions?
In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.
The most virtuosic pages in Scholze's ...
- 119
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
1
vote
0
answers
215
views
Non-noetherian coherent local rings
Let $A$ be a non-neotherian UFD such that $A$ satisfies the following condition $(\sharp)$$\colon$
$(\sharp)$ For any prime ideal ${\frak P}$ of $A$ such that ${\mathrm{ht}}({\frak P}) < \infty$, ...
- 781
20
votes
2
answers
3k
views
Is there a choice-free proof that a Euclidean domain is a UFD?
I asked this question about a week ago on math.SE, without any answers. My motivation is pedagogical, but maybe the question comes closer to research-level than I thought.
The proof (at least the ...
- 11.4k
2
votes
1
answer
216
views
Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme
Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ ...
- 7,609
1
vote
0
answers
189
views
What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]
Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...
- 255
19
votes
1
answer
2k
views
Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?
Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners will be ...
- 3,793
3
votes
2
answers
549
views
Primary structures in $\mathbb Q$
I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
- 7,500
0
votes
0
answers
197
views
Name of some commutative ring akin to $p$-adics
I need help in identifying the naming convention of some commutative ring described below.
Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...
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
0
votes
0
answers
166
views
For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?
For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo $...
- 403
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
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
5
votes
1
answer
415
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
$R^{\...
- 51
1
vote
0
answers
148
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): $...
- 87
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
1
vote
0
answers
342
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 $\mathbb{...
- 446
18
votes
5
answers
2k
views
Is a complete homogeneous symmetric polynomial irreducible?
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 }...
- 446
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
5
votes
4
answers
388
views
Familiar equations in more general settings
What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:
...
- 3,612
36
votes
4
answers
5k
views
What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
- 10.5k