48 votes

Is there a "purely algebraic" proof of the finiteness of the class number?

Yes, there exist purely algebraic conditions on a Dedekind domain which hold for all rings of integers in global fields and which imply that the class group is finite. For a finite quotient domain $A$ ...
A Stasinski's user avatar
  • 3,743
8 votes

Quadratic equation over a global field of characteristic 2

$b/a^2$ has to be of the form $z^2+z$ for $z\in F$. In order for this to happen, it necessary and sufficient (because $F$ is a rational function field) that $b/a^2$ be of the form $z^2+z$ in every ...
Felipe Voloch's user avatar
8 votes

Reference request. Finiteness of the Selmer group

The paper is Milne, J. S. Elements of order p in the Tate-Šafarevič group. Bull. London Math. Soc. 2 (1970), 293–296. He deduces his statement about the Tate-Shafarevich group from a statement about ...
anon's user avatar
  • 141
6 votes

The second Tate-Shafarevich group of a permutation module is trivial

We write $G_w={\rm Gal}(L_w/K_v)$. Definition. For $n\ge 1$, we denote $$Ш_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod_C H^n(C,M)\Big)$$ where $C$ runs over the cyclic subgroups of $G$. Remark. $Ш^2(L/...
Mikhail Borovoi's user avatar
5 votes

Rings of $S$-integers are finitely generated as rings

$\newcommand{\order}{\mathcal{O}} \newcommand{\Z}{\mathbb{Z}} $Here is a short proof, assuming that the class group is torsion (a result for which you should easily find a reference). First, $\order =...
Aurel's user avatar
  • 4,878
5 votes

Is there a "purely algebraic" proof of the finiteness of the class number?

The standard "purely algebraic proofs" of finiteness of class number in the number field case do not in fact proceed by a general method which also works verbatim in the function field case: ...
KConrad's user avatar
  • 49.5k
5 votes

Why is $K_{\upsilon}|K$ separable for a global field $K$?

Since there is already an answer, I want to give my slightly different answer in a special case: Say $K = k(t)$ where $k$ is a finite field and $K_v = k((t))$. We only need to check that $$ K_v \...
Count Dracula's user avatar
5 votes

Why is $K_{\upsilon}|K$ separable for a global field $K$?

By definition an extension of fields $K'/K$ is separable when $K' \otimes_K F$ is reduced for all field extensions $F/K$, and by limit considerations it is the same to say that all finitely generated ...
4 votes

Reference for: Every local field can be realized as the completion of a global field

F. Lorenz: Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics, Theorem 2 p. 78.
Francesco Polizzi's user avatar
3 votes

Does a $K_{\upsilon}$-point of a variety $V$ give a point of $V$ in $K_{(\upsilon)}=K^{sep}\cap K_{\upsilon}$ for a global field $K$?

Let $A$ be a henselian discrete valuation ring and let $A \subset B$ be an extension of discrete valuation rings which has ramification index $1$, induces a trivial extension of residue fields, and a ...
Lucifer's user avatar
  • 31
2 votes

Dedekind criterion for function fields

First note that $M = \overline{\mathbb F_p} (x)$ because that field contains $t$ (it's $f/x^i$), so contains $\overline{\mathbb F_p}(t)$, and is generated over it by $x$, which is a root of $f -t x^i=...
Will Sawin's user avatar
  • 133k
1 vote

Does a $K_{\upsilon}$-point of a variety $V$ give a point of $V$ in $K_{(\upsilon)}=K^{sep}\cap K_{\upsilon}$ for a global field $K$?

If one is prepared to invoke some big theorems, these three situations can all be understood simultaneously in the language of model theory. We say a field extension $E/F$ is elementary just when $E$ ...
Alexander Betts's user avatar

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