# Questions tagged [function-fields]

This tag is for questions related to function field, a finitely generated field extension.

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What is the meaning of the polynomial is 'absolutely additive?' More precisely, let $k$ be a field with characteristic $p$ and $\overline{k}$ be its algebraic closure. Goss' book 'basic structure of ...
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### A proof of existence of canonical divisors

I am confused with the proof of Lemma 1.5.10 in Algebraic Function Fields and Codes by Henning Stichtenoth. Let $0\ne\omega\in\Omega_F$. Then there is a uniquely determined divisor $W\in M(\omega)$ ...
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### Residue fields at points on $\mathbb{A}^n$

Let $k=\bar k$ be a field. I'm trying to "write down" the residue fields at various points on $\mathbb{A}^n=\operatorname{Spec} k[x_1,\cdots, x_n]$, but am having some trouble with the non-...
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### $F/\mathbb{C}(x)$ ramifies at $2$ places $\Rightarrow F=\mathbb{C}(y)$ with $y^n=\frac{ax+b}{cx+d}$

Let $F/\mathbb{C}$ be a function field such that $z\in F$ is integral over $\mathbb{C}\Leftrightarrow z\in\mathbb{C}$. Suppose there is $x\in F\setminus\mathbb{C}$ such that $\mathbb{C}(x)/\mathbb{C}$ ...
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### $\mathcal{O}_S$ holomorphy ring, $S$ containing almost all places $\Rightarrow O_S=K[x_1,…,x_r]$

This is exercise $3.5$ from Stichtenoth's Algebraic Function Fields and Codes: Let $F/K$ be a function field in one variable and $\mathbb{P}_F$ the set of its places. If $S\subsetneq \mathbb{P}_F$ is ...
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### For every function field $F/K$ there exists $x\in F$ with $F/K(x)$ separable

This is exercise $3.2$ from Stichtenoth's Algebraic Function Fields and Codes: Let $F/K$ be function field in one variable such that $K$ is perfect and $\overline{K}\cap F=K$. Given $P_1,...,P_r$ ...
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### If $F'/K'$ is an algebraic extension of $F/K$, then $K'/K$ is algebraic

Let $F'/K'$ and $F/K$ be function fields in one variable such that $K,K'$ are the full constant fields of $F,F'$ respectively. I'm trying to understand the following statement from lemma $3.1.2$ in ...
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### Exact sequence of Riemann-Roch spaces and adeles

I am studying Riemann-Roch theorem on the book by Stitchtenoth, "Algebraic function fields and codes". I am having some troubles understanding a exact sequence of linear mappings between Riemann-Roch ...
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### Definition of a valuation on the points of an elliptic curves

I have to give a presentation about function fields and valuation. I will discuss the case of rational function on the projective line. Then I would like to give an example of a different function ...
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### Is the polynomial degree (difference) a valuation?

Let $k$ be a field and $K=k(x)=\{f/g\mid f, g\in k[x], g\ne 0\}$ the field of rational functions. Let $\mathcal{O}$ be the ring $$\mathcal{O}=\{\frac{f}{g}\in K\mid \deg f\ge\deg g \}\cup\{0\}$$ Then ...
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### Factorizarion criterium to decide whether $K$ is the full constant field of $F/K$

I took the following statement from an answer in math.stackexchange: If $G(x,y)$ is an irreducible polynomial in $K[x,y]$, and $F=K(x,y)$ is an algebraic extension of $K(x)$ defined by $G(x,y)=0$, ...
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### Why assume $K$ is the full constant field of $F|K$?

In the beginning section $1.4$ from Stichtenoth's Algebraic Function Fields and Codes, the author says: "From here on, $F/K$ will always denote an algebraic function field of one variable such that ...
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### Witt Vectors over $\mathbb{F}_q[T]$

I have seen an article where the author constructs a ring of Witt vectors over $\mathbb{F}_q[T]$ with respect to a prime polynomial $P$. That is, it constructs the map (\mathbb{F}_q[T]/P)^{\mathbb{N}...
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### Where can I find this paper by Iwasawa?

I'm trying to find a copy of the 1969 article Analogies between number fields and function fields by Iwasawa. There is a similar paper by Wiles and Mazur titled Analogies between function fields and ...
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### What is the genus of an unramified extension of a function field?

Suppose L/K is a degree $n$m separable, geometric extensions of function fields where both function fields have constants $\mathbb{F}_q$. Suppose also that $L$ is unramified over $K$. Let $g_L$ be the ...
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### Genus behavior under base extension

Suppose that $F$ is a function field of a smooth geometrically connected curve $C$ over $\mathbb{F}_q$. So $F$ is some function field with constants $\mathbb{F}_q$. We could instead look at $C$ over ...
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### how to split a separable algebra?

I'm trying to factor ideals in a function field (more precisely, ideals in a maximal order of a function field), and I've come across a step in the published Buchman-Lenstra algorithm which works in ...