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Function fields isomorphism

Determining if two number fields are isomorphic is a hard problem (Cohen, A course in computational algebraic number theory). Is determining if two functional fields are isomorphic a hard problem? Is ...
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75 views

Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
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1answer
67 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
16 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
106 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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21 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
15 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?
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1answer
37 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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31 views

A question to lemma 4.14 chapter VII “ An Invitation to Arithemtic Geometry” Lorenzini

Lemma 4.14: (constant field extension) Let $k$ be perfect field. Let $k(X)/k$ be a function field. If $k'/k$ is any extension of $k$ in $\bar{k}$, then $k'(X)/k'$ is a function field. Moreover, if ...
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0answers
13 views

Determine the valuation of $\rho$ with $F = k(x,\rho)$ at the only place above $(\infty)$.

Let us assume that we have the following setup. Let $F=k(x,\rho)$ be an algebraic function field with $$f(x,y) = y^n+a_1y^{n-1}+\cdots+ a_iy^{n-i}+\cdots+a_n \in k[x][y]$$ irreducible in $y$, and ...
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34 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
100 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
44 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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38 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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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
71 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
65 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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46 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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41 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 ...
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1answer
41 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 ...
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1answer
64 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
93 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
108 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
278 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
102 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
211 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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94 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
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 ...
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1answer
93 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
292 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
305 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 ...
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1answer
233 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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286 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
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1answer
100 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 ...
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1answer
138 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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176 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 $E$?
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2answers
335 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
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1answer
92 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
281 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 ...
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1answer
126 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 ...
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1answer
177 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 ...