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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
3
votes
Forms over finite fields and Chevalley's theorem
Perhaps it is obvious to most readers, but about a year ago I spent several days trying to determine for which pairs (d,n) there existed an anisotropic degree d form in n variables over a finite field …
- 64.4k
7
votes
Diophantine equation of first degree
It sounds to me like the OP is asking about the Diophantine Problem of Frobenius. This is as follows: let $(a_1,\ldots,a_n)$ be positive integers which generate the unit ideal (i.e., their setwise gc …
- 64.4k
10
votes
prime ideal factorization in an extension field
The best explicit criterion that I know is the criterion of Kummer-Dedekind, which involves writing $K = \mathbb{Q}[t]/(P(t))$ and factoring $P(t)$ modulo the prime $p$. Then the factorization of $(p …
- 64.4k
12
votes
Accepted
Fermat over Number Fields
This is mostly an amplification of Kevin Buzzard's comment.
You ask about points on the Fermat curve $F_n: X^n + Y^n = Z^n$ with values in a number field $K$.
First note that since the equation is …
- 64.4k
6
votes
Question on the abel map and modular parametrization
This is an answer to the question A. Pacetti asked in his comment to Emerton's answer.
The modular variety $A_f$ does not have to be geometrically simple. William Stein and I computed many examples …
- 64.4k
5
votes
Accepted
Nonnegative polynomial in two variables
The following theorem of Artin -- his solution of Hilbert's 17th problem, but in a stronger form than Hilbert himself asked for -- answers the question.
Theorem (Artin, 1927): Let $F$ be a subfield o …
- 64.4k
15
votes
Primes P such that ((P-1)/2)!=1 mod P
The following is a relevant classical paper:
Mordell, L. J.
The congruence $(p-1/2)!\equiv ±1$ $({\rm mod}$ $p)$.
Amer. Math. Monthly 68 1961 145--146.
http://alpha.math.uga.edu/~pete/Mordell61.pdf
…
- 64.4k
14
votes
Accepted
Is there a standard way to read the Legendre symbol?
I say "a on b" for the Legendre/Jacobi/Kronecker symbol. This works because, as an American, I say "a over b" for an ordinary fraction.
- 64.4k
9
votes
Accepted
Do finite places of a number field also correspond to embeddings?
The Archimedean places of a number field K do not quite correspond to the embeddings of K into $\mathbb{C}$: there are exactly $d = [K:\mathbb{Q}]$ of the latter, whereas there are
$r_1 + r_2$ Archim …
- 64.4k
29
votes
Accepted
Number fields with same discriminant and regulator?
Yes, see e.g. the paper "Arithmetically equivalent number fields of small degree" (Google for it) by Bosma and de Smit.
In brief: two number fields $K$ and $K'$ are said to be arithmetically equiva …
- 64.4k
6
votes
Maximal subfields in a division algebra over a local field
To address 2.: for any central simple algebra $A$ over a field $k$, there is a well-developed theory describing the relations between finite splitting fields $l/k$ for $A$ and fields which are sub-$k$ …
- 64.4k
18
votes
What objects do the cusps of Modular curve classify?
Yes, the moduli problem extends to the cusps by way of generalized elliptic curves, i.e., certain semistable curves of arithmetic genus one. For instance, with no level structure there is one point a …
- 64.4k
12
votes
Complete discrete valuation rings with residue field ℤ/p
The classification of CDVRs with residue field any given perfect field k is discussed in Chapter 2 of Serre's Local Fields. In particular:
Theorem II.2: Let R be a CDVR with residue field k. Suppos …
- 64.4k
10
votes
CM of elliptic curves
Allow me to say something which is not so much an answer to this question as to a (very natural) question that I sense is coming in the future.
There are two possible pitfalls in the definition of "h …
- 64.4k
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
5
votes
What is the history of the name "Chinese remainder theorem"?
The theorem is attributed to the mathematician Sun Tzu, also known as Sun Zi, and not to be confused with the military strategist Sun Tzu (of Art of War fame). People think Sun Tzu lived circa 400 AD …
- 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
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
10
votes
What are Santilli's isonumbers?
I looked at Jiang's monograph for a little while last night. Here is what I could get from it (I am now quoting from memory, so my terminology and notation may not be exactly the same). If $F$ is a …
- 64.4k