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Results tagged with algebraic-number-theory
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user 35416
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
14
votes
Heuristics of Cohen-Lenstra-Martinet
The answer to the question as stated is "no". The original Cohen—Lenstra—Martinet heuristics say nothing about the $2$-Sylow subgroup of the class group of a quadratic field.
These heuristics were lat …
- 13k
3
votes
When is $K(\sqrt{a}, \sqrt{b})$ Galois over $\mathbb{Q}$ for $K$ a cyclic cubic field?
A necessary and sufficient condition is that $a$, $b$ be linearly independent in $K^{\times}/K^{\times 2}$ and the Galois group of $K$ cyclically permute the classes of $a$, $b$, and $ab$ in $K^{\time …
- 13k
4
votes
Accepted
Dihedral extension unramified at primes dividing order of group?
$\DeclareMathOperator\Gal{Gal}$The answer is "yes", and this is an easy exercise in class field theory: if, for example, $q$ is a prime number that is $1\pmod n$, and $F$ is a quadratic number field i …
- 13k
14
votes
Accepted
Finite order elements of $\mathrm{GL}_d(\mathbb{Z})$ that are conjugate to powers of themselves
The answer is "no" in general. There may be an elementary way of seeing this, but I will frame this in representation theoretic terms and will describe a general construction.
The question is equivale …
- 13k
10
votes
How to compute Hilbert class field of $\Bbb Q(\zeta_{31})$?
This is a fun question, and I had already been thinking of making some comments on this in the question you link to. Apologies in advance for the long post.
Actually, the quadratic subfield of $\mathb …
- 13k
19
votes
Accepted
how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?
The cyclotomic field $K=\mathbb{Q}(\zeta_{23})$ contains the quadratic field $F=\mathbb{Q}(\sqrt{-23})$, and $F$ has class number $3$ (it is the first quadratic field, when those are ordered by absolu …
- 13k
3
votes
Accepted
How can we justify the use of Example 5,4 (of Cohen, Lenstra) assuming their heuristics
$\DeclareMathOperator\Aut{Aut}$I infer from the context that the precise meaning of "a random $\mathbb{Z}[\zeta_d]$-module modulo a random principal ideal" means that you start by producing a random $ …
- 13k
16
votes
Accepted
Is there any conditions on a finite abelian group so that it cannot be class group of any nu...
It follows from the Cohen-Lenstra heuristic that every finite abelian group is expected to be isomorphic to infinitely many class groups of real quadratic fields (even to a positive proportion of real …
- 13k
3
votes
Accepted
Local factors determine Weil representations - proof of the cyclic case
Why is the map ${\rm Gal}(FL/L)\to {\rm Gal}(F/K)$ injective?
Elements of ${\rm Gal}(FL/L)$ are automorphisms of $FL$ that act trivially on $L$. To be in the kernel of the above map means to also act …
- 13k
7
votes
Accepted
Does the unit index divide the degree of an extension of number fields?
No, not necessarily. For example, you can take $K\neq \mathbb{Q}$ to be totally real, and take $L$ to be a quadratic CM extension of $K$ whose unit group is equal to that of $K$ (e.g. make sure that $ …
- 13k
5
votes
Accepted
Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?
You would usually want the principal Arakelov divisors, i.e. those of the form $(\sum_{\mathfrak{p}}{\rm ord}_{\mathfrak{p}}(a), \sum_\sigma -\log|\sigma(a)|)$ for $a\in F^\times$, to be cocompact in …
- 13k
13
votes
Fermat's last theorem over larger fields
There might well be an elementary construction of infinitely many points (which I cannot think of right now), but in any case, I think that there are experts out there who expect there to be infinitel …
- 13k
12
votes
On a minimal algebraic number field which satisfies the principal ideal theorem
The answer to your first question is "no". In general, if $K/k$ is a cyclic unramified Galois extension of odd order, then the order of the capitulation kernel (the subgroup of the class group of $k$ …
- 13k
8
votes
Accepted
What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?
The group $\Gamma$ is indeed isomorphic to the Galois groups of the fields in the family, whose class groups one studies. The class groups come with a natural action of $\Gamma$, but under this action …
- 13k
45
votes
Accepted
Has Fermat's Last Theorem per se been used?
Corollary 3.17 in this paper of Stefan Keil uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with …
- 13k