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Questions tagged [function-fields]

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Equivalence between valuations

let $k$ be a finite field and $K=k[t]$ be the function field in one variable. Show that a non-trivial, non-Archimedean absolute value $\|\cdot\|$ on K is equivalent to $|\cdot|_{\mathbb{P}}$ for some ...
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Reference request: proof about result of torsion in Jacobian varieties

I've seen the following result a few times, and I am trying to find a proof of the following fact in a book, paper, or notes. Let $\mathcal{C}$ be a genus $g$ curve defined over $\mathbb{F}_q$. ...
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1answer
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A question about Integral Closures from Hartshorne Chapter 2 Exercise 6.4

The exercise that I'm having trouble with is the following. Hartshorne II.6.4: Let $k$ be a field of characteristic $\neq 2$. Let $f \in k[x_1, \dots x_n]$ be a square free nonconstant polynomial, i....
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Estimate for the genus of a global function field

Let $q$ be a prime power and let $F$ be a global function field, i.e. a finite extension of $\mathbb F_q(t)$. Suppose that $\mathbb F_q$ is its full constant field. Suppose in addition that $F$ has ...
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22 views

Smallest infinite place of a rational function field

Let $f$ be a separable irreducible polynomial of degree $n$ with $f \in F[x]$, where $F$ is a function field. I try to understand the following part of an article: ..., we use the minimum infinite ...
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(Inverse) Completion Mapping

Let $\mathbb{F}_q(t)$ be the rational function field and $\mathfrak{p}_\infty$ the place at infinity. I know that every element $z \in \widehat{\mathbb{F}_q(t)}$ (the completion of $\mathbb{F}_q(...
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31 views

Hensel's lemma for the completion of $\mathbb{F}_q(t)$

If we want to find the roots of a polynomial $f(x)$ modulo a prime $p$ to the power of $n$, we can use Hensel's lemma. Let's say we want to find all roots of $x^3+x^2+4x+1$ mod $49$. Then we can use ...
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Computing the genus of the function field $y^l=(x-a_1)^{n_1}\cdots (x-a_m)^{n_m}$

The following is a question from Rosen's "Number Theory in Function Fields"- Let $l$ be a prime not equal to $char(F)$ and $K=F(x,y)$ a function field where $x$ and $y^l=(x-a_1)^{n_1}\cdots (x-a_m)...
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1answer
36 views

Meromorphic Functions on Riemann Surfaces

My question refers to a step in the proof of Prop. 3.3.5 Szamuely and Tamásin's "Galois groups and fundamental groups": Here the statement and Thm 3.3.3 & lemma 3.3.6: The main ingredients for ...
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1answer
22 views

Computing the zeta function of curves over finite fields.

Given a nonsingular curve $X/\mathbb{F}_q$, how can one efficiently compute its zeta function $Z(X/\mathbb{F},T)$? My current strategy is to determine the genus $g$ of $X$, then count the points $N_1,...
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Finding all primes above $x-a_i$ in the function field $y^2=(x-a_1)\cdots (x-a_n)$

This is a problem from Rosen's "Number Theory in Function Fields". Let $K=F(x,y)$ be a function field, such that $y^2=(x-a_1)\cdots (x-a_n)$, and all the elements $a_i$ are distinct. In $F(x)$ we have ...
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38 views

Unramified primes inside $\mathbb{F}_q(t)$

I have some questions about the following statement: Let $P \subset \mathbb{F}_q(t)$ be a prime, with $q=p^e$ for a prime number $p$ and $f$ a polynomial of degree $n$ with coefficients inside $\...
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1answer
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Let $K/F$ be a function field of genus $g\geq 2$, and $\deg(P)=1$. If $0\leq k\leq 2g - 2$ show there are $g$ values of $k$ s.t $l(kP) = l((k+1)P)$.

Let $K/F$ be a function field. First some notation: $D_K$ is the group of divisors of $K$ (i.e. the free abelian group generated by the primes of $K$). For $x \in K^{\times}$, $(x) = \sum_P ord_P(...
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1answer
105 views

Field Extension over the Field of Rational Functions is Finite

Let $K$ be a field and $K(X)$ the field of rational functions with one variable over $K$. If $T(X) \in K(X)$ show that the field extension $K(X)/K(T)$ is finite. The definition of the field of ...
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Connection between definitions in function fields and on curves

(Sorry for the weird title, I really don't know how to describe this question in a line) I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all ...
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1answer
58 views

$\mathbb{F}_q(t)$ vs $\mathbb{F}_q[t]$?

I am having certain troubles understanding certain manuscript. What is the difference between $$ \mathbb{F}_q(t) \quad \text{ and } \quad \mathbb{F}_q[t]? $$
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Commutators and abelianisations of congruence subgroups in function fields

Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite field, and let $r>1$ be an integer. I'm currently looking for the abelianisation of the congruence subgroup ...
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Showing that a certain prime is a Weierstrass point

This problem is taken from the exercises in "Number Theory in Function Fields" by M. Rosen, chapter 6, page 76: Suppose that $\omega\in \Omega_K (0)$ and has a zero $P$ of degree 1, and that $...
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1answer
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How to find prime elements

Let $K$ be an arbitrary number field and $\mathcal{O}_K$ its ring of integers. I have seen many concrete examples about finding prime elements. For example I calculated the prime elements of $\mathbb{...
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ramification at infinity of a Galois Extension.

Let $\mathbb F_q$ be a finite field, let $f\in \mathbb F_q[x]$, and let $t$ be trascendental over $\mathbb F_q$. Consider the splitting field $M$ of $f-t$ over $\mathbb F_q(t)$. Let $P_\infty$ be the ...
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1answer
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Stuck on a problem in Rosen's book on number theory in function fields.

If $p$ is a prime number and $P \in \mathbb{F}_p[T] = A$ is irreducible, show $(A/ \langle P^2\rangle)^*$ is cyclic if and only if $\deg(P) = 1$. I'm stuck on the $(\implies)$ direction. Let $(A/ \...
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Are finite extensions of global function fields always simple? [duplicate]

A global function field is a finite extension of $\mathbb{F}_p(T)$ for some prime number $p$. A finite field extension $L/K$ is called simple if it has a primitive element, i.e. there exists an $x \in ...
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Function field in one variable over a finite field.

Let $K$ be a finite extension over $\mathbb F_p(t)$. How to prove that $K$ is isomorphic to a finite separable extension of $\mathbb F_p(u)$ for some $u\in K$? If I take $K=\mathbb F_p(t)$, then I ...
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0answers
57 views

How to compute function field of a curve

What does $\bar{F}_q(C)$ (function field of a curve) mean? And, how can I compute it when I know C and $F_q$ (or $\bar{F}_q$)? (especially for q=2)
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Ideal norm well defined on ray class group

Let K be a global function field, L an extension of K and $N_{L|K}$ the divisor norm given by $N(P)=p^{f(P|p)}$ for prime divisors and extended linearly. Denote by $\mathcal{D}_{K}$ the divisor group ...
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1answer
132 views

A sufficient and necessary condition for $\mathbb{C}(f(x),g(x))=\mathbb{C}(x)$?

Let $f=f(x),g=g(x) \in \mathbb{C}[x]$. Is there a sufficient and necessary condition for $\mathbb{C}(f(x),g(x))=\mathbb{C}(x)$? This paper is perhaps relevant, although it deals with polynomials ...
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1answer
120 views

Field of constants of a splitting field

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$, and $f\in \mathbb F_q[x]$. How would one check if the field of constants of the splitting field of $f-t$ over $\mathbb F_q(t)$ ...
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What is the analogy between the plane and function fields?

This question arised when I was thinking about the Lonely Runner Conjecture. The LRC is an example of a Conjecture which we can 'rewrite' in terms of function fields, where we use that $\mathbb{Z}$ is ...
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1answer
111 views

The Galois group of a specialized polynomial

Let $P(t,x)\in\mathbb Q[t,x]$ be irreducible with Galois group $G$ over $\mathbb Q(t)$. It is known that if $t_0\in\mathbb Q$ is such that $P(t_0,x)$ is separable, then the Galois group of this ...
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1answer
37 views

Looking for specific irreducible inseperable polynomial over $\mathbb{F}_2(t)$ where t an indeterminate.

I have the field $\mathbb{F}_2(t)$, where $\mathbb{F}_2$ is the finite field, and $t$ an indeterminate. Now $X^2+t$ is an irreducible inseperable polymomial, and $X^4+t$ also. I was looking for an ...
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38 views

Non zero discriminant

Let $p$ be a prime number. $k=\bar F_p$. Let $G=g(t)+F[t,x]\in k[t][x]$ be such that $G$ is irreducible in $x$ and separable in $t$, where $x$ is a polynomial in $F_{p}[t]$ of the form $f=\sum_{i=0}^{...
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Irreducibility in Function Fields

I was reading a paper on Function Fields. There is a statement like this. " Let $F \in F_q[t][x]$ here $x$ is a variable over $F_q[t]$and $deg_{x} F>0$ and $q$ is a power of $p$. Here author ...
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Rational point of variety over function fields

I am currently study Field Arithmetic written by M.Fried and M.Jarden. Here is one of the theorem as following which is Propostion 13.4.6 in that book: Every field K has a regular extension F ...
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1answer
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Show $f(x)$ is irreducible Let $f(x)=x^4+x^2+t \in \mathbb{F}_2 (t)[x]$

Show $f(x)$ is irreducible Let $$f(x)=x^4+x^2+t \in \mathbb{F}_2 (t)[x]$$ let $y=t$ $$f=y+x^2(x^4+1)y^0 $$ let $p=x^2$ use Eisensteins criterion. basing it out of an example from hungerford ...
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how to check for irreducibility

This might be a simple question, but I'm stuck. What is meant by $f(x,y)=ax^2+bxy+cy^2$ is a polynomial over $F_{p^l}(t)$., which is a function finite field, how to check for irreducibility? Does ...
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Degree of the field extension $\bar{k}(Z_f) /\bar{k}(x)$

I came across an argument on the lecture notes. It does not have a proof of the fact and I want to understand the argument. Let $k$ be a field, $f(x,y) \in \bar{k}[x,y]$. Define $Z_f := \{(a,b) \in ...
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Why is the degree of a rational map of projective curves equal to the degree of the homogeneous polynomials?

Let $C_1 \subseteq \mathbb{P}^m$ and $C_2 \subseteq \mathbb{P}^n$ be projective curves, and let $\phi : C_1 \rightarrow C_2$ be a nonconstant rational map given by $\phi = \left[ f_1, \ldots, f_n \...
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1answer
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Exercises reference in function field theory.

I am currently trying to learn about algebraic function fields. I am using as a reference Algebraic function fields and codes from Henning Stichtenoth. This book is unfortunately rather difficult for ...
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QR algorithm convergence for fields other than R and C

I was wondering how the QR algorithm could be used to calculate the eigenvalues of matrices containing elements which are members of fields other than R and C. For example, say we have a matrix of ...
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1answer
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Is it true that if there exists a rational place then $K$ is algebraically closed in $F$?

I am working through "Algebraic function fields and codes-Henning Stichtenoth" (2nd Edition) and I got a little confused about the following: First of all an algebraic function field $F/K$ of one ...
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1answer
179 views

An homomorphism between the function fields of integral schemes induces a morphism between schemes

Following Hartshorne, AG, Chapter 1, Theorem 4.4, I am trying to prove that given a field homomorphism between the function fields of two integral scheme $\phi:k(Y)\longrightarrow k(X)$ there exists a ...
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If a morphism of curves induces Galois extension of function fields, does the galois group act transitively on the fibers?

Let $f:X\to Y$ be a non-constant morphism of smooth projective curves defined over an algebraically closed field. Suppose that the corresponding field extension $K(X) \setminus K(Y)$ is Galois. Does $\...
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1answer
87 views

Birational morphisms of surfaces

I read that in the category of projective smooth models of a function field $K = k(x,y)$ ($k$ algebraically closed), there is at most one morphism between two such models (the morphism being a ...
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1answer
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Inverse element in the function field of an affine variety

First of all I want to mention all the definitions of our lecture, so there will be no misunderstanding: Let $X \subset \mathbb{A}_k^n$ be an affine variety, meaning $X$ is irreducible and closed ...
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228 views

Primitive elements of finite fields

Let $p$ be a prime number and $q=p^n$ for some positive integer $n$. $F_q[x]$ is the polynomial ring with coefficients in $F_q$. For any $M(x)\in F_q[x]$, define $\mathcal{R}(M(x))\subset F_q[x]$ to ...
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1answer
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When can an algebraic number be approximated by a $p$-adic number?

Let $F$ be an algebraic function field in one variable over the finite field $\mathbb{F}_{p}$. In particular, $F$ is not perfect. Let $a \in F-F^p$ and $$f(Y)=Y^p - a \in F[Y]$$ be a purely ...
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1answer
78 views

Generating element of local ring in a purely inseparable extension

Let $F$ be an algebraic function field (in one variable) over the finite field $\mathbb{F}_{p}$. Let $F'$ be a purely inseparable extension of $F$ of degree $p$. In particular, $F'$ is a simple ...
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1answer
152 views

Why does genus 0 imply that the function field has one generator?

I have read that given a genus 0 modular curve $X$ defined over a base field $k$. The corresponding function field k(X) can be generated by one modular function. I have also read that converse, that ...
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27 views

Degree in Finite Extensions of $\mathbb{F}_q(X)$

Let $F\in\mathbb{F}_q[X]$. I.e. a polynomial with coefficients in $\mathbb{F}_q$. Then there is an obvious way to define the degree: the highest power of $x$ with a non-zero coefficient. There are ...
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Characterizing elements with same (function field) norm knowing principal divisors

Let $L$ be a hyperelliptic function field ($[L : K(x)]=2$), with $K$ an algebraically closed field. The norm is defined as $N(f)=f\bar{f}$, where $\bar{f}$ is the conjugate of $f$. If $y^2+h_1(x)=h(x)$...