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Results tagged with algebraic-number-theory
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user 37644
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
20
votes
1
answer
2k
views
When complex conjugation lies in the center of a Galois group
Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question concer …
- 1,499
4
votes
How much do I need to learn algebraic geometry to understand arithmetics over number fields
I'm going to say no: it's not necessary to know the modern algebraic geometry to explore the theory of rational points of algebraic varieties. Obviously, as pointed out in some of the answers, it's h …
- 1,499
4
votes
2
answers
489
views
The best possible density in Hilbert's Irreducibility Theorem
Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/ …
- 1,499
4
votes
0
answers
834
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), 43–44 …
- 1,499
3
votes
Is the infimum of Salem numbers > 1?
Just wanted to add: the conjecture that Salem numbers are bounded away from one (the "Salem Conjecture") is wide open as the other answers state, and it's equivalent to part of what is known as the "S …
- 1,499
2
votes
0
answers
165
views
algorithm to find a new point of small height in a number field extension
By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$
If $K$ is a number field, let $\delta(K)$ denote …
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