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20 views

extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is $v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...
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0answers
33 views

The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$

If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and ...
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0answers
18 views

Generalisation of Fermat's Little Theorem in Function Fields

There is a well-known generalisation of Fermat's Little Theorem as for any $a\in\mathbb{Z}$ and $n\in\mathbb{N}$ we have $$\sum_{d\mid n} \mu(n/d) a^d\equiv 0\pmod n.$$ where $\mu$ is the Möbius ...
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2answers
32 views

Finite extensions of $\mathbb F_p(t)$ [on hold]

Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$. Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some ...
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1answer
60 views

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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0answers
19 views

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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0answers
39 views

Endomorphism ring of Drinfeld modules.

Let $\mathcal{X}$ be a smooth geometrically irreducible projective curve over $\mathbb{F}_q$. Fix a closed point $\infty\in \mathcal{X}(\bar{\mathbb{F}_q})$. Let $K$ be the function field of ...
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0answers
4 views

does the $\zeta_K$ function of a function field determine the genus of that function field?

Let $K_1$ and $K_2$ both be function fields over a finite field (or algebraic curves, if you like) with zeta functions $\zeta_{K_1}$ and $\zeta_{K_2}$. Say that $\zeta_{K_1} = \zeta_{K_2}$ - so that ...
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0answers
18 views

kernel of the artin map when dealing with S-ideles and S-divisors for function fields

Let $L/K$ be an abelian extension of function fields and let $\vartheta_{L/K}:\mathcal{D} \to \text{Gal}(L/K)$ be the Artin map from the divisors of $K$ to the galois group. What can be said about ...
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1answer
39 views

Relation between ramification locus of a tower and of it's constant field extension

I am trying to understand Remark 7.2.22 (Page 256) of Algebraic Function Fields and Codes (Second Edition) by Henning Stichtenoth. In that remark he considers a tower $\mathcal{F} = ...
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0answers
28 views

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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0answers
13 views

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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1answer
57 views

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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0answers
39 views

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 ...
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0answers
20 views

Primes Splitting in the Gaussian Integers - Function Field Analogue

The function field analogue of the ring $\mathbb{Z}[i]$ are functions of the form $A(T)+\alpha B(T)$, where $\alpha$ is a solution to the equation $x^{2}+T=0$, over $F_{q}[T]$. We know that a prime ...
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0answers
38 views

How computationally difficult is it to determine if two function fields are isomorphic?

Determining if two number fields are isomorphic is a hard problem, according to Cohen in his book A Course in Computational Algebraic Number Theory. Is determining if two function fields are ...
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1answer
72 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring ...
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1answer
19 views

function field of open curve and compactificaton

Let $U$ be a smooth curve over some field $k$ and $C$ the only smooth projective curve containing $U$ as a dense open subset. Can someone help me understading why the function fields of $U$ and $C$ ...
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1answer
129 views

poles and zeros of function field of $\mathbb{P}^1$.

In which condition: an element of function field of $\mathbb{P}^1$ has zero or pole or no-zero&no-pole. I am thinking that: since $\mathbb{P}^1$ and $\mathbb{A}^1$ is birrationally equivalent ...
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0answers
26 views

Representation Matrix computation time estimation

Abstract/general version of the question If we have a matrix $$ M_z = \begin{pmatrix} \sum_{j=1}^{n}\lambda_j \lambda_{j,1,1} &\dots & \sum_{j=1}^{n}\lambda_j \lambda_{j,n,1}\\ ...
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0answers
24 views

Product of monic polynomials in finite fields

I am trying to show that the product of monic polynomials of degree $n$ in $\mathbb{F}_p[T]$ is given by $\prod_{i=0}^{n}(T^{p^n}-T^{p^i})$. I tried generating function but with no luck. Any hint?
3
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1answer
62 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field of one variable $x$ where $$y^2 = f(x) \in K[x].$$ We want to compute the full constant field of $F$ (i.e. ...
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0answers
50 views

Unramified Galois extension of a function field of a curve - definition

Reading some papers I've encountered the following: Let $X$ be a smooth curve over a field $K$ with function field $K(X)$. Consider the maximal Galois extension of $K(X)$ which is unramified over $X$. ...
3
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1answer
117 views

When a holomorphy ring is a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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0answers
52 views

When is a holomorphy ring a PID? [duplicate]

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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0answers
54 views

valuation on function fields

Let $p$ be an irreducible polynomial in $k[x]$, for some characteristic 0 field $k$. So we have a valuation $v_p$ corresponding to $p$. Now take $a$ to be a root of $p(x)$. Then $(x-a)$ is irreducible ...
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0answers
43 views

genus of the field $\mathbb{R}(x,y)$ where $x^2+y^2+1=0$

I have come across a problem while reading Chevelley's "Introduction to the theory of algebraic functions of one variable", in which he says that the genus of the field $L=\mathbb{R}(x,y)$ is 0, where ...
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1answer
69 views

Explicit examples of higher genus Drinfeld modules?

Let $C$ is a smooth, projective, geometrically irreducible curve over $\mathbb{F}_q$ and $\infty$ a closed point of $C$. A Drinfel'd module over $A = H^0(C, \mathcal{O}_C)$ is an injective ...
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2answers
84 views

$\dim (D-P)=\dim (D)-1$

I'm trying to prove this question: Let $D$ be a divisor in $F|K$ such that $\dim (D)\gt 0$ and $0 \neq f\in \mathscr L(D)$. Thus $f\notin \mathscr L(D-P)$ for almost all $P$. Then show that ...
3
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0answers
76 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field ...
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0answers
54 views

Function field of a projective variety

I am reading Silverman's "The Arithmetic of Elliptic Curves". On page 10 he defines the function field of a projective variety $V$ over a field $K$ to be the function field of $V\cap\mathbb{A}^n$, ...
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1answer
177 views

Is the following a field?

I think I may need a refresher in logs here. The question is: F=$\{a \in R \vert a<1\} 1<t \in R$ (1)$a\#b= a+b-ab$ for all $a,b \in$ F (2)$a*b=1-t^{log_t(1-a) * log_t(1-b)}$ for all $a,b ...
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0answers
50 views

Function fields for genus 2 curves

Let's say you were given a genus one algebraic curve by the equation $y^2 = (x-a)(x-b)(x-c)$ and you wanted to parametrize it. We could go ahead and convert it to Weierstrass form: $y^2 = 4t^3 - g_2t ...
3
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1answer
54 views

Geometric meaning of the splitting field over a function field

Let $K$ be a field and consider the ring of polynomial in two variables $K[x,t]$. Now take a polynomial $f(x)\in K[x]$ of positive degree and consider it in the bigger ring $K(t)[x]$. Suppose that ...
3
votes
1answer
78 views

Alternative models of a hyperelliptic curve

I am studying some particular examples of hyperelliptic curves and their automorphism groups (from this paper, if it is of interest). It is mentioned in the paper to understand the automorphism of ...
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1answer
123 views

Product of monic polynomials in a Function Field

I'm trying to prove that given $P$ is a monic irreducible over $\Bbb F_q[t]$ and $d=\deg(P)$, then $$\prod_{\substack{f\text{ monic}\\ 0\leq\deg(f)<d}}f=\pm1\pmod{P}.$$My first thought goes to ...
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0answers
159 views

Why is every Divisor of the rational function field $K(x)$ over $K$ a principal divisor if $K$ is algebraically closed

We have the rational function field $K(x)$ and the algebraically closed $K$. One of the statements in my lessons was: Then every Divisor $D \in $Div$(K(x)|K)$ is a principal divisor, so there exists ...
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2answers
323 views

Metric completion of field of fractions

The integers have as a field of fractions the rational numbers which have a metric completion as the real numbers. The reals can be represented by infinite decimal expansions which can be approximated ...
3
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1answer
112 views

what is the constant field of irreducible components a divisor?

Let $D$ be a divisor on an algebraic variety over a field $k$, that is $$ D=\sum n_i D_i $$ where $D_i$ are the irreducible components. I came across the expression "the constant field of $D_i$" and ...
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1answer
282 views

extension of algebraic function field

Let $K$ be a field, $t$ a transcendetal element over $K$ and $F'|K(t)$ an infinite Galois extension. Hence I have a tower of extension $K\subset K(t) \subset F' $. Does exist a subextension $F$ of ...
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0answers
102 views

places of function field and closed point of a scheme

Given an integral scheme $X$, let $K(X)=\mathrm{Frac}(R)$ be its function field, where $\mathrm{Spec}(R)$ is some non-empty open affine subscheme of $X$. Take the maximal ideal $P$ of some DVR of ...
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1answer
89 views

Why do number rings have no endomorphisms

This question is about the analogy between number fields and function fields. It's a soft question and the title misrepresents the question. Consider the projective line over a field. This has many ...
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1answer
94 views

Levels of Rings and Fields, -1 as a sum of squares

Definition: Let $R$ be a commutative ring. The level of $R$, denoted $s(R)$, is the least positive integer $s$ such that $-1$ can be written as the sum of $s$ many squares in $R$. Set $s(R)=\infty$ if ...
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1answer
351 views

Completion of a rational function field w.r.t. place at infinity

When taking the completion a rational function field, say $k(t)$, with respect to the place at infinity, most books refer to this using the notation $k((1/t))$. Since $k((t)) = k((1/t))$ (EDIT: this ...
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1answer
381 views

Two notions of uniformizer

Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
2
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1answer
251 views

Hyper-elliptic curves in positive characteristic

I have been looking at hyperelliptic curves in characteristic two, in particular using Algebraic Geometry and Arithmetic Curves by Qing Liu, which gives a description in all characteristics. For the ...
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0answers
313 views

Artin-Schreier extensions over characteristic two fields

I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do ...
2
votes
1answer
129 views

Residue map of a place

The following definition for a residue map is given in "Algebraic Curves over a Finite Field" by Hirschfeld, Korchmáros and Torres (page 265): "Let $\Sigma$ be a field of transcendence degree 1 over ...
3
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1answer
142 views

A field isomorphism related to polynomial rings and their field of fractions

There are 2 ways to approach function fields: the algebraic approach, i.e. looking at finite extensions of $K(s)$, where $s$ is transcendental. The other is geometric, i.e. considering functions over ...
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0answers
204 views

Separability of compositum of fields

Let $E/F$ be a finite separable extension, and let $K$ be a function field with constant field $F$. Is the compositum $KE$ of $K$ and $E$ a separable extension over $E$?