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Results tagged with algebraic-number-theory
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user 26522
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
19
votes
Accepted
Multizeta function values
The elements of $S$ are conjectured to be $\mathbb{Q}$-linearly independent, and so a basis for the $\mathbb{Q}$-linear span of the multiple zeta values.
This is what Francis Brown accomplished at t …
- 13.8k
16
votes
Special topics to include in course in algebraic number theory
Here are a few suggestions, though within number theory.
Chebotarev's original proof of the density theorem (in the logarithmic form). Or at least the infinitude of totally split primes as well as t …
- 13.8k
14
votes
3
answers
925
views
Asymptotics for algebraic numbers of height less than one
The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$?
The rather cr …
- 13.8k
14
votes
0
answers
641
views
Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?
Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields unram …
- 13.8k
13
votes
Estimate number of solutions in the Roth's theorem
For a fixed $\alpha$, the number $N_{\alpha}(\epsilon)$ is bounded by a polynomial function of $1/\epsilon$. The proof of this requires either Faltings's product theorem, or Esnault and Viehweg's mult …
- 13.8k
11
votes
2
answers
1k
views
Can there be a power basis for a totally real field of high degree?
A number field $K$ is said to have a power basis if there is an $\alpha \in K$ such that the full ring of integers $O_K$ is the $\mathbb{Z}$-linear span of $1,\alpha,\alpha^2,\ldots,\alpha^{\deg{K}-1} …
- 13.8k
9
votes
0
answers
263
views
How small may the discriminant of an $S_d$-field be?
In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree …
- 13.8k
9
votes
Accepted
Is the infimum of Salem numbers > 1?
I believe it is the general opinion, at least among those working in diophantine approximations, that the extreme case of Salem numbers (the question of the title) would be just as difficult as the fu …
- 13.8k
7
votes
Accepted
Finite Nontrivial Unramified Towers of Number Fields
It is certainly possible that $1 < [L:F] < \infty$, i.e. that the extension $F^{\mathrm{un}}/F$ be finite and non-trivial. The simplest example of this is $F = \mathbb{Q}(\sqrt{-5})$. Its Hilbert clas …
- 13.8k
4
votes
Asymptotics for algebraic numbers of height less than one
As I indicated above, $\log{N(d)} \asymp d^2$ reduces to a problem about irreducible polynomials, the very likely affirmative answer to which would prove the lower bound $n(h,d) \geq (h-o(1))d^2$, as …
- 13.8k
4
votes
Old question of Serre on discriminants of a sequence of polynomials
To reiterate Serre's question (Open Problem 2.5 in Odlyzko's survey): Must there be only finitely many polynomials having root discriminant below a given bound?
With this answer I just want to note …
- 13.8k
3
votes
0
answers
100
views
Independence of number fields generated by roots of Littlewood polynomials
Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and
$$
c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^ …
- 13.8k