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Results tagged with nt.number-theory
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user 3272
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
8
votes
Accepted
Bateman-Horn, continued even further
See the following two papers of Nobushige Kurokawa, both appearing in Proc. Japan Acad. Ser. A:
"On Some Euler Products II" (volume 60, 1984, 365-368, esp. Proposition 1)
"Special Values of Euler Pr …
- 49.1k
19
votes
Accepted
Class number of non-maximal order in imaginary quadratic function field?
There is a formula that works in all degrees, not just imaginary quadratic. In a global field $K$, let $O$ be integral over ${\mathbf Z}$ or ${\mathbf F}[t]$ (${\mathbf F}$ a finite field) and be "bi …
- 49.1k
4
votes
Accepted
Reference needed for Lucas' Theorem for multinomial coefficients modulo a prime
You can prove this by induction on the maximum number of base $p$ digits, and to make the argument simpler it's better to formulate a mildly stronger theorem where the leading base $p$ "digit" is allo …
- 49.1k
2
votes
The norm of a non-Galois extension of local fields
Hi Erick. Yes, this is true. Maybe it's in Iwasawa's Local Class Field Theory? For the global analogue, see the last exercise in Cassels & Frohlich.
- 49.1k
21
votes
Accepted
Independence over Q and Kroneckers result
This is on wikipedia. See Kronecker's theorem. It was proved by Kronecker in 1884.
The necessary and sufficient condition for integral multiples of a point $(r_1,\dots,r_n)$ in the $n$-torus $(\mat …
- 49.1k
5
votes
Accepted
D K Faddeev's construction of quaternionic fields
I don't have the paper you refer to, but on the page http://www.math.spbu.ru/vestnik/2008/vestnik0801/dfaddeev.pdf, which is dedicated to the 100th birthday of Faddeev, his early work is described in …
- 49.1k
1
vote
A question about non-archimedean binomial expansion
What do you mean in the comment to your post that you “get confirmation of $s=1$ from other people”? In any case, it is “simple" to show $s=1$ if you are familiar enough with binomial expansions in $p …
- 49.1k
10
votes
Techniques for computing fundamental units in cubic extensions
Use Artin's inequality: if $K$ is a cubic field with one real embedding and $v > 1$ is a unit in $O_K$ then $|\text{disc}(K)| < 4v^3 + 24$. (I use inequalities on elements of $K$ via the one real embe …
- 49.1k
12
votes
Accepted
Quotients of a ring of integers
The answer to your main question is no. Let $I = (p)$ where $p$ splits completely in $L$ and $r := [L:\mathbf Q] > p$. (Example: cubic field in which $p = 2$ splits completely.) Then $\mathcal O_L/(p) …
- 49.1k
10
votes
Cases where the number field case and the function field (with positive characteristic) are ...
I answered this question once already, but another example where $\mathbf Z$ and $k[x]$ behave differently when $k$ is a finite field was brought to my attention recently by Jeff Lagarias and it deser …
- 49.1k
31
votes
Primes P such that ((P-1)/2)!=1 mod P
There is some history to this question. Dirichlet observed
(see p. 275 of ``History of the Theory of Numbers,'' Vol. 1) that since we already know
$(\frac{p-1}{2})! \equiv \pm 1 \bmod p$,
computing …
- 49.1k
9
votes
Accepted
Connection between Möbius and 𝜔 function
Yes, that deduction is possible, but somewhat indirectly.
The estimate $\sum_{n \leq x} \mu(n) = o(x)$ is equivalent to the nonvanishing of $\zeta(s)$ on ${\rm Re}(s) = 1$ since both conditions are kn …
- 49.1k
10
votes
Squares in a triquadratic field
Set $K = \mathbf Q(\sqrt{-3},\sqrt{5},\sqrt{-7})$ and $\alpha = 1170\sqrt{-3}\sqrt{5}\sqrt{-7}-19110$.
I think the "best" way to show $\alpha$ is not a square in $K$ is not to show something else is n …
- 49.1k
21
votes
Field complete with respect to inequivalent absolute values
For each prime $p$, the field $\mathbf C_p$ is isomorphic to the field $\mathbf C$ as abstract fields (i.e., they are isomorphic purely algebraically, ignoring absolute values on them) since all algeb …
- 49.1k
50
votes
Accepted
Is there a ring of integers except for Z, such that every extension of it is ramified?
Yes, there are examples and Minkowski's proof for ${\mathbf Q}$ can be adapted to find a few of them. Some examples of this kind among quadratic fields $F$, listed in increasing size of discriminant …
- 49.1k