# Tag Info

Accepted

### 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 $A$ ...
• 3,743

### 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 ...
• 30.2k
Accepted

### 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 ...
• 141
Accepted

• 4,878

### 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: ...
• 49.5k

### 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 \...
• 1,376
Accepted

### 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 ...
Accepted

### 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.

### 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 ...
• 31