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Results tagged with algebraic-number-theory
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user 1050
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
15
votes
2-torsion in class groups of cubic fields
I'm happy to announce a new result of the shape you ask for: if $F$ is a cubic field (of any signature) then the size of the 2-torsion in its class group is bounded above by $O(D_F^{0.2785})$. The sam …
- 7,337
6
votes
Exercise in Milne's CFT notes
I think Milne is right.
If a prime $p$ does not ramify in fields $K_i$, then it does not ramify in their compositum. So you know that 5 can't ramify in $L$, because it visibly doesn't ramify in any …
- 7,337
18
votes
3
answers
2k
views
Did Hermite really prove "Hermite's Theorem" on number field discriminants?
Hermite's theorem, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant.
The usual proof (see Neukirch's Algebraic Number Theory …
- 7,337
2
votes
Old question of Serre on discriminants of a sequence of polynomials
You can certainly choose the polynomials $P_n(t)$ so that your sequence is unbounded: there are polynomials of every degree of arbitrarily large discriminant.
You can also choose them so that your se …
- 7,337
9
votes
Accepted
Has anyone found an error in an early version of Neukirch?
In my edition of Neukirch, Chapter I.9, Exercise 2:
If $L|K$ is a Galois extension of algebraic number fields, and $\mathfrak{P}$ a prime ideal which is unramified over $K$ (i.e. $\mathfrak{p} = \mat …
5
votes
Accepted
polynomials with the same discriminant
[Substantial edit: As I mentioned previously, cubic fields can have the same discriminant but not be isomorphic; but I've revised my answer to better address the author's question.]
There are a lot of …
- 7,337