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Results tagged with algebraic-number-theory
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user 4213
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
2
votes
Is there a purely algebraic criterion which characterizes the real algebraic numbers?
As other correspondents have pointed out there is no algebraic way
to distinguish the elements of $\mathbb{Q}^{alg}\cap\mathbb{R}$ but there
is an algebraic way of distinguishing totally real algebrai …
- 20.8k
31
votes
Which number fields are monogenic? and related questions
To add to Keith's answer, there are various classes of number fields which are
known to be not monogenic. For instance, the following paper
Marie-Nicole Gras,
Non monogénéité de l'anneau des entiers …
- 20.8k
1
vote
What are Mean Values of Ideal Densities in Galois Extensions?
This may be a reference to the the Davenport-Heilbronn theorem
on the distribution of discriminants of number fields.
See
math.stanford.edu/~fthorne/davenport-heilbronn.pdf
for a nice exposition. Str …
- 20.8k
5
votes
Is there a notion of Galois extension for Z / p^2?
There's the notion of Galois ring. Let $K$ be the degree $m$
unramified extension of $\mathbb{Q}_p$
and let $\mathcal{O}_K$ be its ring of integers. Then the quotient
$R=\mathcal{O}_K/p^n\mathcal{O}_ …
- 20.8k
20
votes
$A_5$-extension of number fields unramified everywhere
Here's the standard example. I found it in Lang's Algebraic Number Theory
where he attributes it to Artin. Let $K$ be the splitting field of $X^5-X+1$
over $\mathbb{Q}$. Then $K$ has Galois group $S_5 …
- 20.8k