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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