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45 views

When is a holomorphy ring 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
39 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
34 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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0answers
17 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 ...
3
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2answers
69 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 ...
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0answers
56 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
34 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
173 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
34 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
35 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
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1answer
56 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 ...
4
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1answer
85 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 ...
0
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0answers
93 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
255 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
92 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
187 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
86 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 ...
2
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1answer
85 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 ...
4
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1answer
88 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 ...
2
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1answer
250 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 ...
3
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0answers
56 views

Algebraic Curves similar to Hyper-Elliptic Curves

Throughout, $F_q$ will denote a finite field of $q$ elements with characteristic $p \neq 2$. It is well-known that the equation $y^2 = f(x)$ (for square-free $f \in F_q[X]$) defines an hyper-elliptic ...
2
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1answer
266 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
227 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
269 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
82 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
130 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 ...
1
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0answers
160 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 $E$?
2
votes
2answers
302 views

Existence of morphism of curves such that field extension degree > any possible ramification?

Throughout I would like to work over an algebraically closed field of characteristic 0 (so no separability issues), say $k$. My question is the following: Do there exist two curves $X$ and $Y$ and a ...
2
votes
1answer
86 views

Linear disjointness of two “explicit” field extensions

Let $k$ be a characteristic zero field and let $L/k$ be a quadratic extension. Write $L = k(\sqrt{p})$. Let $q$ be a non-square in $k^\star$ and let $r \in k^\star$ be any constant. Consider the ...
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1answer
263 views

Is a field perfect iff the primitive element theorem holds for all extensions, and what about function fields

Let $L/K$ be a finite separable extension of fields. Then we have the primitive element theorem, i.e., there exists an $x$ in $L$ such that $L=K(x)$. In particular, the primitive element theorem ...
3
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1answer
125 views

On Intermediate Fields of $\mathbb{C}(x_1,\dots,x_n)$

I am recently reading some Galois Theory, and a question occurred to me: What are the intermediate fields of $K$ of $\mathbb C(x_1,\dots,x_n)$, where $n$ is an arbitrary integer? I am aware of a ...
9
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1answer
174 views

Another Tangent on Tangents

This question asked yesterday got me thinking. While the derivatives of the tangent function span an infinite dimensional vector space over $\mathbb{C},$ the transcendence degree of the field ...