Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with algebraic-number-theory
Search options not deleted
user 14830
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
16
votes
Accepted
Positive primes represented by indefinite binary quadratic form
Class field theory promises such a polynomial (more properly,
such a number field $H$, since a polynomial generating $H$ might have to err
on the first few primes, though in our case it turns out ther …
- 79.9k
15
votes
Accepted
Special units in the $11$th cyclotomic field
Yes. Indeed
$$
(1 + \zeta + \zeta^{10}) \, (\zeta + \zeta^4 + \zeta^7 + \zeta^{10}) = 1
$$
with $\sum_i a_i = 3$ and $\sum_i b_i = 4$; now change each $a_i$ to $a_i+3$
and each $b_i$ to $b_i+3$. ( …
- 79.9k
6
votes
Special units in the $11$th cyclotomic field
If I did this right there are a total of $1045 = 55 \cdot 19$
solutions, obtained from the following $19$ basic solutions by changing
$a_i,b_i$ to $a_{ri+s\bmod 11}$ and $b_{ri+s \bmod 10}$ for all
$s …
- 79.9k
12
votes
Accepted
How small can a totally positive integer be?
$\alpha_n$ can be exponentially small once $M$ is large enough,
say $M \geq 6$.
For $m > 0$ let $\tau_m$ be the (monic, degree-$m$) polynomial
such that $\tau_m(z+z^{-1}) = z^m + z^{-m}$; in other wo …
- 79.9k
6
votes
Field of minimum discriminant of a given degree
That's already false for $n=4$, when the field of smallest $|D|$ has dihedral Galois group, and indeed of the ten smallest $|D|$ only two come from fields with Galois group $S_4$. [Explicitly, $f(4) …
- 79.9k
2
votes
Linear independence of algebraic integers of equal norm
A cubic counterexample: Let $K$ be the first totally real cubic field
${\bf Q}[x] / (x^3+x^2-2x-1)$ (the roots are $2 \cos (2m\pi/7)$ for $m=1,2,3$);
and let $a=13$, the first totally split prime in …
- 79.9k
5
votes
Accepted
Linear independence of algebraic integers of equal norm
We adapt an idea from a now-deleted answer by Kenny Lau
to construct examples for any $n>2$ with the $\alpha_j$ all contained in
2-dimensional space. Let $a$ be prime, and choose distinct integers
$x …
- 79.9k
8
votes
Accepted
Fundamental units of imaginary quartic fields
There's certainly some uniform bound, as a special case of the
theorem that for each $n$ and $M$ there are only finitely many
algebraic integers $\epsilon$ of degree $n$ each of whose conjugates
has a …
- 79.9k
11
votes
Accepted
The existence of infinitely many supersingular primes for every elliptic curve over Q
It's the auxiliary prime $l$ that must be $3 \bmod 4$;
the supersingular prime $p$ is not guaranteed to be congruent to $3 \bmod 4$,
and indeed the residue of $p \bmod 4$ is unpredictable
(unless the …
- 79.9k
13
votes
Accepted
Numbers integrally represented by a ternary cubic form
Your conjectures are correct. So was the "someone else at MSRI [who]
muttered something about norm forms" (mentioned in earlier edits
of the question), except for the part about laughing at you.
As …
- 79.9k
5
votes
The missing Euler Idoneal numbers
The reference
Weinberger, P. J.: Exponents of the class groups of complex quadratic fields, Acta
Arith. 22 (1973), 117–124.
was what Matthias Schütt and I cited for the fact that "there is at most …
- 79.9k
18
votes
Accepted
A family of polynomials with symmetric galois group
[Edited mostly to incorporate references etc. from Michael Zieve]
The polynomial $(x^n-1)/(x-1) - y$ has Galois group $S_{n-1}$ over ${\bf C}(y)$ for each $n$, as expected. This answers one of the t …
- 79.9k
11
votes
Accepted
Can elliptic integral singular values generate cubic polynomials with integer coefficients?
The flurry of comments did not yet produce an answer to the question
concerning the complete elliptic integrals
$$
I_1 := \frac12
\int\limits_0^\infty\dfrac{dt}{\sqrt{t(t+1)(t+\frac{8+3\sqrt{7}}{16})} …
- 79.9k
9
votes
Accepted
Square-free grows as $6n/\pi^2$: $k$-th free?
Yes, it works in much the same way for any $k$. Here's an elementary proof.
Let $Q_k(n)$ be the number of $k$-th power free integers $\leq n$.
Then
$$
Q_k(n) = \sum_{d^k \leq n} \mu(d) \lfloor n/d^k …
- 79.9k
7
votes
Accepted
The existence of elliptic curves with prescribed supersingular primes
No. In fact the set can be arbitrarily sparse, i.e. the $n$-th prime
in the set can be chosen to exceed $a_n$ for any sequence $\{a_n\}$.
This is because the rationals are countable. Fix an enumerat …
- 79.9k