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Results tagged with algebraic-number-theory
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user 50081
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
9
votes
2
answers
2k
views
Completion and algebraic closure
Following this question:
Given a valued field $K$, denote with $\bar{K}$ its algebraic closure and with $\hat{K}$ the completion. Then both $\hat{\bar{K}}$ and $\hat{\bar{\hat{K}}}$ are complete and …
- 1,101
3
votes
2
answers
751
views
Primes of the form $p=x^2-ny^2$
I want to know which primes $p$ can be written in the form $p=x^2-ny^2$ for given $n \in \mathbb{N}$. If $n$ is squarefree, $n \not\equiv 1 \mod 4$ and $\mathbb{Z}[\sqrt{n}]$ is a principal ideal doma …
- 1,101
0
votes
0
answers
701
views
Extensions of $\mathbb{Z}[\sqrt{-n}]$ that are UFD
EDIT: Since the original question was to vague I will pose some stronger conditions:
Given $n \in \mathbb{N}$ with $n \geq 3$ I want to know if there is a ring $R$ with the following properties:
$\ …
- 1,101