# Tagged Questions

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### Geometric equivalent of the degree zero divisor class group of an algebraic function field (in the singular case)

In an algebraic function field $F/k$ we have the degree zero divisor class group $\text{Cl}^0(F/k)$. Now since any such function field corresponds to the function field of a normal projective curve ...
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Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is: $$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$ where $v$ ranges ...
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### Quotients of Elliptic Curves

I am fairly inexperienced with elliptic curves so there might be aspects of my question that may need better wording but let me know if there are any issues: Question: Say I have an elliptic curve ...
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### $A^\times/k^\times$ is a free $\mathbb{Z}$-module of rank of at most $r - 1$

Consider an algebraically closed field $k$, a finite extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, the integral closure $A'$ of $k[1/T]$ in $K$, and the integral closure $A''$ of ...
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### Characters on rings of residue classes modulo polynomials over finite fields

First recall the following orthogonality relation on $\mathbb{Z}/n\mathbb{Z}$. Fix $n \in \mathbb{Z}$, $n \neq 0$. For $r \in \mathbb{Q}$, let $e(r) := e^{2 \pi i r}$. Let $x \in \mathbb{Z}$. Then ...
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### Finding Primitive Elements of Separable Function Field Extensions

Suppose you have a a curve $C$ defined by an equation in $x$ and $y$. There is a map from $C$ to $\mathbb{P}_1$ by projection onto $x$. This corresponds to a separable extension of function fields ...
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### Is $\mathbb{C}(x,y)$ a rational function field?

Let $\mathbb{C}(x,y)$ be a degree $2$ extension of $\mathbb{C}(x)$ where $y$ is a root of $p(Z)=Z^2 + (x^2+1)$. Is it true that $\mathbb{C}(x,y)$ is not a rational function field? In other words, ...
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### A remark about Eisenstein Criterion in Stichtenoth's Algebraic Function Fields and Codes

I am reading Stichtenoth's Algebraic Function Fields and Codes and am confused about a remark in Chapter 3. He mentions that Proposition 3.1.15, which proves irreducibility of a certain kind of ...
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### How does Galois theory over function fields compare to that of number fields?

Let $M(S)$ be the function field of a surface. We can then consider field extensions, auto morphia groups, ... How does Galois theory in this setting compare to that of number fields? What plays the ...
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### Compute $\ell(W+P)$ and $\ell(W-P)$ for a place $P\in\mathbb{P}(F/K)$ of degree $1$ and any $W$ canonical divisor.

Let $F/K$ be a function field of genus $g$. Let $P\in\mathbb{P}(F/K)$ be a place of degree $1$ and $W$ be any canonical divisor. Determine $\ell(W+P)$ and $\ell(W-P)$. If $\ell(P)=\deg(P)+1$, then we ...
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### profinite completion of a ring of integers in a global field

It is well known that the profinite completion of the integers, $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ is, by the Chinese remainder theorem, isomorphic to $\prod_p \mathbb{Z}_p$. I ...
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### Cokernel of map, function field.

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places of $F$, and let $S$ be a nonempty finite subset of $X$. We are interested in the dimension ...