# Questions tagged [function-fields]

This tag is for questions related to function field, a finitely generated field extension of transcendence degree $n>0$ of a field of constants $k$.

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### Properties of field extensions of finite transcendence degree

Let $K/F$ be a field extension of transcendence degree $n\ge 2$. If $E$ is a subfield of $K$ such that $F\subset E \subset K$ and the transcendence degree of $E$ over $F$ equals $n-1$, then $K$ is a ...
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### Why is (y) a uniformizer for $y^2 = x^3 + x$ at (0,0) and why is its order equal to 1?

I'm currently trying to understand Silvermans example for the valuation on curves discussed in the answer to this post: Definition and example of "order of a function at a point of a curve" ...
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1 vote
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### On the complexity of global fields isomorphism

Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
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1 vote
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### Why is the Function Field of the Modular Curve $X(N)$ defined over $\mathbb{Q}(\mu_n)$?

Following Section 7.6 in Diamond & Shurman, the algebraic model of $X(N)$ is constructed (a priori) over $\mathbb{Q}$ by first defining its function field as \begin{equation*} \mathbb{Q}(j,f_{(0,1)...
1 vote
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### Relation Between Dimension of Rational AG Code and Degree of Associated Place

I'm working through Henning Stichtenoth's Algebraic Function Fields and Codes as part research for my undegraduate dissertation, and I'm unsure of the proof of a particular proposition (proposition 2....
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### $n$-torsion parts of rank 2 Drinfeld modules and elliptic curves over function fields

I'm studying Drinfeld modules to study elliptic curves over function fields. (We assume the characteristic $p$ is large enough if we need.) Simply, we consider rank $2$ Drinfeld $A$-module $\rho$ over ...
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### Possible typo in Artin-Schreier generator, Stichtenoth's book

In the book "Algebraic function fields and codes" by H. Stichtenoth, pg. 332 he treats cyclic-p Galois extensions $L/K$ in positive characteristic and says: An element $\gamma_1\in L$ such ...
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### Show that local rings on a curve are PID

I'm reading the notes on Elliptic Curves from this MIT course, more specifically this part where the local ring of a curve $C$ at point $P$ is defined, as the set of rational functions $f$ on $C$ such ...
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### Riemann Roch over finite fields according to Weil

In the somewhat old (but extremely nice) reference Adeles and Algebraic Groups by Weil, he cites Riemann-Roch as part of an argument for computing the volume of $\mathbb{A}_k/k$ (paraphrased slightly):...
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### for arbitrary $y∈\Bbb{F}_q((X))^{sep}$, there exists $c∈\Bbb{F}_q((X))$ and $a∈\Bbb{F}_q((X))^{sep}$ such that $y＝ca^q$

$\Bbb{F}_q((X))^{sep}$ be separable closure of $\Bbb{F}_q((X))$. I want to prove, for arbitrary $y∈\Bbb{F}_q((X))^{sep}$, there exists $c∈\Bbb{F}_q((X))$ and $a∈\Bbb{F}_q((X))^{sep}$ such that $y＝ca^q$...
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### Irreducibility of varieties after base change and finite extentions of pure transcendental fields

I've met a curious question when I did the exercise on page 20 of Serre's Topics in Galois Theory. Let $K$ be a field of characteristic $0$, $W \to \mathbb{P}^1_K$ be a finite dominant morphism of ...
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