1
vote
0answers
153 views

$\mathfrak{q}$-ideal class bound

Let $K$ be a number field, $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{q}$ be a nonzero ideal in $\mathcal{O}_K$. The $\mathfrak{q}$-ideal class group consists of equivalence classes of ...
8
votes
1answer
223 views

Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?

It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...
1
vote
2answers
178 views

Double Density Theorem?

A colleague asks me the following: "I wonder if you can give me a reference - or a guidance where to look – from a fact I recall from graduate school. I’m sure it can be generalized quite a bit but ...
0
votes
0answers
82 views

Number fields of given degree and bounded discriminant (Paper request)

I have been searching several Libraries to find the following important paper of Wolfgang Schmidt. "Schmidt, Wolfgang M.(1-CO), Number fields of given degree and bounded discriminant. (English ...
3
votes
2answers
303 views

Cohen-Lenstra Heuristics reference

I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the ...
9
votes
1answer
429 views

The Dissertation of F. J. van der Linden

Does anyone have access to the 1984 dissertation of Franciscus Jozef van der Linden under Hendrik Lenstra? It is called Euclidean Rings with two infinite primes. The theory is that this has the ...
3
votes
0answers
162 views

Euclidean real quadratic fields

It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic ...
2
votes
0answers
148 views

Lang's preprint “Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure”

I am trying to find the following preprint of Serge Lang, which supposedly discusses his C1 conjecture: "Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure". I have not ...
2
votes
1answer
179 views

RefReq: Algorithms for standard operations in Algebraic Number theory

Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code ...
1
vote
0answers
107 views

Reference request for a basic result on relative differents & discriminants

I am looking for a better reference for the results in this extremely short and elementary paper: Tôyama, Hiraku, `A note on the different of the composed field', Kōdai Math. Sem. Rep. 7 (1955), ...
5
votes
2answers
387 views

On Weil's characters of type (A)

In Weil's paper "On a certain type of characters of the idele-class group of an algebraic number field", Weil introduces a class of characters on the Idele class group (of not necessarily finite ...
4
votes
0answers
163 views

Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?

Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
9
votes
1answer
603 views

The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
5
votes
2answers
391 views

Book on ideal theory in Hurwitz quaternions

Hello, I am looking for a book that studies the set of Hurwitz quaternions (HQ). In particular, I am interested in a connection between HQ and imaginary quadratic fields (IQF); quaternion orders ...
9
votes
1answer
353 views

Which formulae of Euler is Fröhlich referring to?

In A. Fröhlich's article Local Fields in Algebraic Number Theory, the following claim is made: if $R$ is a Dedekind domain with field of fractions $K$, $L$ is a finite separable extension of $K$ and ...
15
votes
3answers
1k views

Commutative Algebra with a View Toward Algebraic Number Theory

Someone asked me this today, and I don't know what the standard answer is: Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...
6
votes
1answer
353 views

Generalization of Hilbert 94 and capitulation

Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
6
votes
2answers
695 views

Decomposition of finite algebras over finite fields

Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
6
votes
1answer
533 views

Where can I find a modern write-up of Heegner's solution of Gauss' class number 1 problem?

In a recent MO question someone mentioned Heegner's solution of the Gauss "class number 1" problem which takes the following form: When the class number of an imaginary quadratic form is 1 an ...
1
vote
1answer
687 views

Good Minkowski Theory and Commutative Algebra Books

I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory. However, I am interested in learning algebraic number theory and I recently found that the ...
7
votes
0answers
411 views

ideal classes in quadratic number fields

Let $m$ be a squarefree odd integer that can be written as a sum of two squares, and let $K = {\mathbb Q}(\sqrt{m}\,)$ be a real quadratic number field with fundamental unit $\varepsilon$. Let $m = ...
6
votes
1answer
361 views

The Jacobian ideal generates the socle of a complete intersection

This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here: http://tinyurl.com/2967eov I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
16
votes
3answers
2k views

Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
8
votes
2answers
880 views

Artin motives, References for.

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
10
votes
6answers
1k views

Reference for Learning Global Class Field Theory Using the Original Analytic Proofs?

Hi Everyone! I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find ...