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Results tagged with nt.number-theory
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user 1149
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
9
votes
Accepted
remark in milne's class field theory notes
The point is that it is one thing to show that two mathematical objects are isomorphic; it is another (stronger) thing to give a particular isomorphism between them. A rather concrete instance of thi …
- 64.4k
2
votes
Accepted
Possible values for differences of primes
Since there are infinitely many primes, the set $K$ is certainly infinite, so in the expression $\frac{|K|}{|\mathbb{Z}^+|}$, you are attempting to divide two infinite cardinalities. This is not a me …
- 64.4k
9
votes
Accepted
Ideal classes and integral similarity
A very belated answer to 1), but: I just saw that this is treated very nicely in Curtis and Reiner's Representation Theory of Finite Groups and Associative Algebras. Theorem 20.6 therein not only wor …
- 64.4k
14
votes
Accepted
Elliptic curves and prime numbers
(Sorry, I misread the question at first.) The following result reduces your question to a problem of analytic number theory:
Theorem (Hasse-Deuring-Waterhouse): For a prime $p$ and $N \geq 1$ the fo …
- 64.4k
13
votes
Accepted
Rational points over completions of a number field
Without loss of generality $X$ is affine, so embed it in projective space and apply the Bertini Theorem to conclude that $X$ contains a smooth, geometrically integral affine curve $C^{\circ}$ missing …
- 64.4k
5
votes
How probable is it that a rational prime will split into principal factors in a Galois numbe...
I believe that Felipe's answer is not correct. [Edit: rather, it is correct according to a different interpretation of the question. But my interpretation is also natural, I think.]
Say a prime $p$ …
- 64.4k
3
votes
Opinions about the book "Lectures on Algebraic Geometry 1: Sheaves, Cohomology of Sheaves, a...
Here is the MathSciNet review by Andrei D. Halanay:
The book under review is the first of a two-volume introduction to algebraic geometry. This first volume deals mostly with prerequisites, namely ho …
4
votes
Accepted
Finiteness of Obstruction to a Local-Global Principle
"Has there been further progress in this area since 1993?"
So far as I know, there has been no direct progress. I feel semi-confident that I would know if there had been a big breakthrough: Mazur wa …
- 64.4k
13
votes
Statements in group theory which imply deep results in number theory
The fundamental theorem of arithmetic (uniqueness of factorization of integers into primes) is an immediate consequence of the Jordan-Holder theorem on uniqueness of composition factors of finite grou …
17
votes
Accepted
Does $\pi_1(Spec(\mathbb{Z}[1/p]))$ depend on p?
The full etale fundamental groups in question are, I believe, complicated infinite profinite groups. (They are however "small" in the technical sense that they have only finitely many open normal sub …
- 64.4k
1
vote
possible CM-types of abelian varieties
I posted the following answer yesterday after only a quick skim of the question. When I read it with more care, it seemed to me to be the answer to a different question entirely. After having looked …
- 64.4k
28
votes
Accepted
Which p-adic numbers are also algebraic?
The field $K_p = \mathbb{Q}_p \cap \overline{\mathbb{Q}}$ is a very natural and well-studied one. I can throw some terminology at you, but I'm not sure exactly what you want to know about it.
1) It …
- 64.4k
35
votes
Accepted
Did Pogorzelski claim to have a proof of Goldbach's Conjecture?
In the 1970's Pogorzelski published a sequence of four papers in Crelle concerning the Goldbach Conjecture (and various generalizations and abstractions):
MR0347566 (50 #69) Pogorzelski, H. A. On the …
- 64.4k
4
votes
The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$
This issue came to my attention for the first (and only, up until now) time when Gregory Dresden gave a talk about resultants of cyclotomic polynomials in the UGA number theory seminar last spring. I …
- 64.4k
17
votes
What do you call this ring?
As David Speyer says, the most common ways of referring to $\prod_p \mathbb{Z}_p$ are "$\mathbb{Z}$-hat" or "the profinite completion of $\mathbb{Z}$''.
However, I have also heard it called "the Pruf …
- 64.4k