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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
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
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
19
votes
Accepted
How to picture $\mathbb{C}_p$?
You do whatever works for you. Some people think more algebraically, others more geometrically. I certainly don't know what "to picture" means in this context, but then, I am a more algebraic person, …
- 13k
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
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
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
5
votes
Examples of DVRs of residue char p and ramification e
Take any polynomial of degree $e$ that is Eisenstein at $p$, adjoin to $\mathbb{Q}_p$ a root of that polynomial and you will get a totally ramified extension of $\mathbb{Q}_p$ of degree $e$. Moreover, …
- 13k
3
votes
Elliptic curves over finite fields
As Xandi Tuni said, most of the answers to your questions can be found in standard references.
Silverman, Knapp's book on elliptic curves, Milne's book, many more (just google for elliptic curves).
…
- 13k
5
votes
What (permutation) groups can occur as galois groups of irreducible polynomials of degree n
This question is ambiguous and can be understood in various ways, since no mention is made of the base field. If you are asking "for what subgroups $G$ of $S_n$ do there exist fields $L,K$ such that $ …
- 13k
4
votes
Examples of Using Class Field Theory
As I have written in your question on SE, if you want to know how to actually compute polynomials that give you ring class fields for a given modulus, then Cohen's Advanced Topics in Computational Num …
- 13k
3
votes
Accepted
Proof of a Simple Converse in Algebraic Number Theory
If you throw in the residue degrees as well, then you get Zev's question. Otherwise, the converse is not true. As an example, consider a quadratic field that has an unramified everywhere $A_5$ Galois …
- 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
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
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
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