Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 …

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x) …

Hello, Some books and courses take the approach to Riemann-Roch for curves considering only the algebraic viewpoint of algebraic extensions of a field k(t) without going into any a …

I'm reading the proof of Lemma 4.1 [1] which says: "Let $F = K(x,y), y^q = f(x)$, where $q$ is a prime different from characteristic of $K$. Let $Z := Gal(F/K(x))$ and we have $Z …

Let $K/F$ be a function field with exact field of constants $F$ ($F$ is a finite field of characteristic $p$ prime). A prime in $K$ is a discrete valuation in $K$ containing $F$. I …

Suppose we have an irreducible polynomial $f\in K[x]$. Is there some way to sometimes tell whether $f$ splits completely after adjoining just one root of $f$ to $K$? I am mostly i …

Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?" There seems to be a general philosophy that problems over function …

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' …

Hello, Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and functi …

Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or usefu …

Are there any restrictions on the ground field over which the implicit function theorem holds? In particular, does the theorem hold over function fields like $F_q((1/t))$?

if $P_{1}$ and $P_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P_{1})=P_{2}$. then, does $\deg (P_{ …

Let $f \in \mathbf{F}_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}_q[T]/(f))^\times$ can be constructed by adjoining torsion points …