All Questions
Tagged with nt.number-theory ac.commutative-algebra
360 questions
7
votes
1
answer
474
views
Fibonacci embedded in Catalan?
Given a partition $\lambda$ and its Young diagram $\pmb{Y}_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}_{\lambda}$. We now ...
- 43.2k
7
votes
1
answer
683
views
An analogy between the ring of polynomials in two variables and another (commutative?) ring
One of the answers to this question says:
"In Serge Lang's Algebra, he says: "One of the most fruitful analogies in mathematics is that between the integers $\mathbb{Z}$ and the ring of polynomials $F[...
- 2,837
6
votes
2
answers
1k
views
Question about the sum of odd powers equation
Quite surprisingly the following question appears while studying the billiard dynamics.
Assume we have $2n$ real numbers: $ x_1, x_2,..., x_{2n}$.
Assume also that $S_k=0$ for any odd positive integer ...
- 728
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
6
votes
1
answer
247
views
Sets of $\mathbb{F}_p$-points of closed subvarieties of $\mathbb{A}^n$
Let $p$ be a prime and let $n\geq 2$ be an integer.
The set $\mathbb{A}^n(\mathbb{F}_p)$ has $p^n$ elements so it has $2^{p^n}$ subsets. How many of those subsets are of the form $V(\mathbb{F}_p)$ ...
- 103
6
votes
1
answer
821
views
Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?
Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.
I have two questions:
Does there exist a discrete valuation subring $R$ of $K((t))$ of ...
- 10.7k
6
votes
1
answer
894
views
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 ...
- 119
6
votes
1
answer
571
views
Original proof of Hilbert irreducibility theorem
Does there exist a modern exposition of Hilbert's original (1892) proof of the Hilbert irreducibility theorem? Of course, I can (and will) read Hilbert's original article, but I would feel more ...
- 1,294
6
votes
1
answer
1k
views
A weak version of Bass' conjecture
Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
- 21.3k
6
votes
1
answer
294
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. ...
- 16.2k
6
votes
1
answer
1k
views
reference for p-local and p-complete integers
Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, ...
- 499
6
votes
2
answers
329
views
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 ...
- 515
6
votes
1
answer
262
views
Is there a finite extension with a non-trivial class group of any PID?
Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?
- 73
6
votes
1
answer
356
views
Constructive Bezout cofactors in the ring of algebraic integers
We see the following in an answer to This Question : (mangled by me for my purposes, all errors my fault)
Dedekind stated (in 1871) that the ring of algebraic integers is a Bezout Domain. He calls it ...
- 30.1k
6
votes
3
answers
990
views
Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?
Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of ...
- 6,792
6
votes
1
answer
411
views
When is $1+a+a^2+\dotsb+a^{{\rm ord}_n(a)-1}$ divisible by $n$?
I posted this question on math.SE 10 days ago and had no answer, comment or vote. If an answer is not available, I could really use a reference point as well.
For the sake of completeness, I am ...
- 111
6
votes
2
answers
563
views
If the discriminant of a binary quadratic form has high valuation, is the form "almost a square".
For a binary quadratic form $ax^2+bxy+cy^2$ over a field (characteristic not 2), the discriminant $b^2-4ac$ is 0 if and only if that form is the square of a linear form. I am curious about an ...
- 2,955
6
votes
1
answer
951
views
Is it possible to recover the degree of a field extension from a list of elements and the ground field?
I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
- 1,049
6
votes
1
answer
242
views
$(q,t)$-Fibonacci polynomials: area & bounce statistics
This is related to my earlier (unanswered) MO post. Preserve notations from there.
We take advantage of the one-to-one correspondence between the $(s,s+1)$-core partitions and $(s,s+1)$-Dyck paths. ...
- 43.2k
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 ...
- 507
6
votes
0
answers
119
views
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
6
votes
0
answers
357
views
Ideals of orders in number fields
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{O}$ be an order of $K$ and $A$ a proper ideal of $\mathcal{O}$, by which I mean that $\mathcal{O} = \{\alpha \in K : \...
6
votes
0
answers
120
views
How does $\mathcal{O}_{C_K} / p \mathcal{O}_{C_K}$ looks like?
I am reading stuff about Fontaine's periods rings. Let $K$ be a $p$-adic field and $C_K = \widehat{\overline{K}}$. Then $\mathcal{O}_{C_K}/p\mathcal{O}_{C_K}$ isn't perfect, otherwise $R = \...
- 133
6
votes
0
answers
106
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. ...
- 96.4k
5
votes
3
answers
1k
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.
$\textbf{Questions:}$
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\cO}{\mathcal{O}}$
...
- 3,555
5
votes
1
answer
551
views
Is decomposability of integer polynomials over the rational numbers an undecidable problem?
By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as
$$
F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]),
$$
which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
- 290
5
votes
2
answers
1k
views
Unramified (finite) extensions of fields complete with respect to a discrete valuation
Hello,
I've been reading the excellent online book on Algebraic Number Theory by J.S.Milne. In the section described above there is a footnote maintaining that the separability of the residue field ...
5
votes
1
answer
209
views
Units of $\mathbf Z[X,Y]/(P(X,Y))$
Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.
What is known about the group of units of $A$?
It's not even clear to me that ...
- 2,521
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
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
1
answer
223
views
Intrinsic characterisation of a class of rings
This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...
- 6,214
5
votes
1
answer
504
views
Classification of the quotients of the ring Z/4 [X]
Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes ...
- 63.1k
5
votes
1
answer
2k
views
Generalizing Dedekind's Factorization Theorem
A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
$...
- 14.6k
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
5
votes
1
answer
2k
views
Irreducibility of some trinomials modulo $p$
Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in
http://www.mscand.dk/article.php?id=1472,
(If the link does not work try googling: ...
- 1,641
5
votes
1
answer
293
views
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
5
votes
1
answer
213
views
Can the strongest Hensel lemma over integer rings imply smoothness over $\mathbb Z_p$?
Let $X \rightarrow \mathbb Z_p$ be a flat finite type morphism, with reduced special fiber and smooth generic fiber.
Assume $X(O_K) \rightarrow X(O_K/m_K)$ is surjective for all fintie extension $K$ ...
- 461
5
votes
2
answers
369
views
Links between tight closure and deformation theory
I am looking for links between tight closure and deformation theory. As a sample question:
Question 1. Are there geometric interpretations in terms of deformation theory of
Frobenius rationality?
...
- 32.1k
5
votes
2
answers
2k
views
Order of vanishing of an integer polynomial at a point
Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless $...
- 7,836
5
votes
1
answer
544
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.
...
- 103
5
votes
0
answers
144
views
Is there a good notion of higher-rank archimedean norm?
Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
- 63.9k
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
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
5
votes
0
answers
148
views
Can we express the degree 10 and degree 15 Galois resolvents of sextic binary forms in terms of its basic invariants?
Let $V_6$ denote the 7 dimensional $\mathbb{C}$-vector space of binary sextic forms. For $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...
- 26.9k
5
votes
0
answers
153
views
On factorization algorithms for $\mathcal{O}[x]$
We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of $\...
user76479
5
votes
0
answers
194
views
How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]
Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...
- 5,637
4
votes
2
answers
367
views
Equality in $\mathbb F_q\left(\left(\frac1T\right)\right)$
Can one characterize the $a\in\mathbb F_q\left(\left(\frac1T\right)\right)$ such that $a(T+1)=a(T)$? Although this seems elementary, I did not manage to find a answer.
Thanks in advance for any help.
- 3,998
4
votes
1
answer
398
views
Does irreducible polynomial remain reduced by pre-composition?
Let $f(x),g(x)$ be polynomials in $\mathbb{Q}[x]$. If $\mathrm{deg}(f)\geq2$ and $f$ irreducible, is the composition $f(g(x))$ always reduced (has no repeated irreducible factors)?
(If we do not ask $\...
user39380
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 ...
- 1,598
4
votes
1
answer
444
views
Surfaces of general type
What methods do we know about proving-disproving existence of rational points on surfaces of general type?
I was recently asked. My gut answer was- 'nothing'.
user95222