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 ...
merev's user avatar
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1 answer
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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" ...
Zedssad's user avatar
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1 vote
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40 views

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 ...
Reyx_0's user avatar
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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)...
Josu P. Z.'s user avatar
1 vote
1 answer
67 views

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....
ouroboring's user avatar
2 votes
0 answers
23 views

$n$-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 ...
jawlang's user avatar
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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 ...
user128787's user avatar
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1 vote
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70 views

Two definitions of places on a curve

In the literature on algebraic curves, I have found two definitions of places on a curve. Let $K$ be an algebraically closed field. Let $\mathcal{C}$ be an irreducible curve defined by $f(x,y) \in K[x,...
kiyopi's user avatar
  • 255
2 votes
0 answers
102 views

Generators of the function field of an elliptic curve

I came across the following claims (from https://aghitza.github.io/publication/translation_velu/, the very beginning of section 1). Let $E$ be an elliptic curve (in Weierstrass form) over over an ...
Myath's user avatar
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3 votes
2 answers
124 views

Fixed field of the linear fractional maps ($X \to \frac{aX+b}{cX+d}$) of $k(X)$ where $k$ is finite.

I come up with this interesting question in Serge Lang's Algebra: Let $k$ be a field of $q$ elements and $K=k(X)$ be the rational field of one variable. Let $G$ be the group of automorphisms obtained ...
Degenerate D's user avatar
1 vote
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Computing the p-rank of Divisor class group for function field

In the context of my work, I am trying to develop an algorithm to factorize some operators on algebraic function fields of positive characteristic $p$. To this end I need to be able to compute ...
raphitek's user avatar
2 votes
1 answer
152 views

Notes on global function fields

Does anyone happen to know a good set of notes on global function fields? In particular, I'm hoping to find something akin to Neukirch's chapter on Riemann-Roch theory, but for global function fields ...
delpsi's user avatar
  • 105
2 votes
1 answer
118 views

Connexion between affine and projective variety defined by a homogeneous polynomial

Let $P\in K[X_1,\dots,X_n]$ be a homogeneous polynomial. It then defines an affine variety $X=V(P)\subset \mathbb{A}^n_K$ like any polynomial, but also a projective variety $Y\subset \mathbb{P}^{n-1}...
Captain Lama's user avatar
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188 views

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 ...
Weier's user avatar
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How to check if $f(X) \in F(g_1(X), \ldots, g_m(X))$?

Let $X=(X_1,\ldots, X_n)$ and $g_1(X), \ldots, g_m(X) \in F(X)$ for a field $F$. Which algorithms or methods are used to determine if $f(X) \in F(g_1(X), \ldots, g_m(X))$ is true for a given $f(X) \in ...
Valentin's user avatar
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1 answer
113 views

Do $L$-functions of varieties over function fields have analytic continuation / functional equations?

Suppose that $K$ is a function field (i.e: a finite extension of $\mathbf{F}_q(t)$) and $X$ is a smooth projective variety over $K$. I have a few questions about the $L$-function of $X/K$: How do you ...
Adithya Chakravarthy's user avatar
1 vote
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47 views

Is it possible to describe extension of places of a function field in MAGMA?

I would like to describe, for example, the unramified places of an extension $F'/F$ of function field (in one variable) in MAGMA calculator. More specifically, given $F'/F$ an extension of function ...
Adler Marques's user avatar
2 votes
0 answers
118 views

A question about the use of Silverman's Specialization Theorem

In the book "The Arithmetic of Elliptic Curves", 2nd Edition, by Silverman, there is a theorem called Silverman's Specialization Theorem (Theorem 20.3 pp. 457) which states that: Let $K$ be ...
the inner beauty's user avatar
4 votes
0 answers
167 views

Traditional and modern approaches to class field theory

I read in Gerald Janusz's book Algebraic Number Fields where he says there are two approaches to class field theory. The traditional approach, as in his book, uses L-series, Dirichlet density and ...
Ja_1941's user avatar
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2 votes
0 answers
59 views

Inverse Galois Characteristic p [closed]

I am trying to find any information on the inverse galois problem for F_p[t]. Is there a literature or survey for this problem or any well known counter examples. Thanks :)
Jacob Lewis's user avatar
2 votes
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66 views

Adelic theta function over function fields

I saw the following claim on Godement-Jacquet's classical book "Zeta functions of simple algebras": (on page 153) Let $F$ be a global function field, $\Phi$ a Schwartz function on $\mathbb{A}...
youknowwho's user avatar
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4 votes
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107 views

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):...
C.D.'s user avatar
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0 answers
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Galois extension of function field with Galois group C_6

Let $p>3$ be a prime number and let $\mathbb{F}_{p}(t)$ be the field of rational functions over $\mathbb{F}_{p}$. Is there a curve $C$ such that the Galois group $\operatorname{Gal}(\mathbb{F}_{p}(...
Steppewolf's user avatar
4 votes
1 answer
145 views

Confusion about the term "infinite prime"

I try to learn more algebraic number theory and I am confused about the terminology "infinite prime" and "finite prime" in global fields. If $K$ is a global field, do we mean by a ...
Daniel W.'s user avatar
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1 vote
1 answer
114 views

What is the $ord_P$ function associated to $R$?

I am busy studying Rosen's "Number Theory in Function Fields" Chapter 5. In this chapter Rosen defines a prime $M$ in the function field $K$ over $F$ to be a discrete valuation ring $R$ with ...
joe's user avatar
  • 157
1 vote
0 answers
157 views

Number of places of given degree in a global function field

Let $q$ be a prime power and $\mathbb{F}_q$ the field with $q$ elements. We know that the places of the global function field $\mathbb{F}_q(x)$ are given by monic, irreducible polynomials in $\mathbb{...
Severin Schraven's user avatar
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1 answer
39 views

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 $y=ca^q$...
Pont's user avatar
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11 votes
1 answer
277 views

An exercise in Galois theory dealing with function fields

Our professor gave an example in class: compute the Galois group of $ x^4-2x^2-1$ over $\mathbb{ Q}$. So the main steps are: $\alpha:=\sqrt{1+\sqrt{2}}$, and observe that the splitting field is $\...
youknowwho's user avatar
  • 1,407
1 vote
1 answer
96 views

Adele space of rational function field

Let $F=K(x)$ and $K(x)/K$ be the rational function field. I am then trying to prove \begin{align*} A_F = A_F(0) + F \end{align*} where $A_F$ is the adele space, $A_F(0)$ the set of adeles $\alpha$ ...
Johan's user avatar
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1 vote
0 answers
30 views

Field generated by two rational functions

Let $K$ be a field (not necessarily infinite) and $K(x)$ be the field of rational functions in the variable $x$. Let $g(x)$ and $h(x)$ be in $K(x)$. For fixed degrees, and "generically", I ...
Reyx_0's user avatar
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2 votes
0 answers
111 views

Galois representation of an elliptic curve over a function field of char p.

Let $E/K$ be a non-isotrivial elliptic curve over a function field of characteristic $p$. Let $$\rho_{E,l}:Gal(K^{sep}/K)\to GL_2(\mathbf{Z}_l)$$ For $l\neq p$, the image of $\rho_{E,l}$ is a $l$-adic ...
user12770's user avatar
  • 171
-1 votes
1 answer
74 views

Prove that the ring $C^0[a,b]$ is not Euclidean

I'm trying to prove that the ring $C^0[a,b] = R$ is not Euclidean. I think I need to use proof by contradiction for that. Suppode there is a Euclidean function $d$ on $R$. Then $\forall a,b \in R, b \...
SpaceNugget's user avatar
1 vote
0 answers
85 views

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 ...
pop1's user avatar
  • 43
0 votes
1 answer
161 views

Confusion about the rank of a valuation

Let $v: K/k\to \Lambda$ be a valuation on a field $K$ extension of $k$, with values in an ordered group $\Lambda.$ It is often claimed that the rank of $\Lambda$ and the rational rank, $\dim_\mathbb Q ...
Adam's user avatar
  • 538
0 votes
0 answers
41 views

Galois group of $s^6 - t - 1$ in $\mathbb{F}_p(t)$

I was wondering how one would determine the galois group of the extension $\mathbb{F}_p(t,s)$ of $\mathbb{F}_p(t)$, where $s^6 = t + 1$? I certainly know (and Magma reassures) that this depends on the ...
Math4Life's user avatar
  • 145
8 votes
0 answers
204 views

Galois group of a function field over finite field

I have a question about the structure of this Galois group that I can't understand: suppose that $p>2$ is prime and $q$ is any power of p, and we have these two function fields: $$K=\mathbb{F}_{p}(...
Soroush Salehin's user avatar
3 votes
0 answers
91 views

Monogenic function fields

Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
Andry's user avatar
  • 73
4 votes
1 answer
119 views

What is the generalization of composition in abstract algebra (of rings and fields)?

I'm newer to abstract algebra, and recently I came back to the concept of a field after finding rational functions form a field. But, what I notice is that rational functions (and algebraic functions) ...
StackQuest's user avatar
0 votes
0 answers
28 views

C-planes and associated valuations in the paper of James Ax titled "On Schanuel's conjecture"

I am currently trying to understand Proposition 2. (page 257 of the JSTOR version) in the above cited paper of Ax. It says the following: Let $F\supseteq C \supseteq \mathbb{Q}$ be a tower of fields. ...
Andry's user avatar
  • 73
2 votes
2 answers
63 views

Does $t^{p-1}$ have an antiderivative in $\Bbb F_q(t)$?

The formal derivative on $\mathbb{F}_p[t]$ is not onto, since $t^{p-1}$ (for instance) has no antiderivative. Does it have one if we extend the formal derivative to $\mathbb{F}_p(t)$? By the quotient ...
anon's user avatar
  • 152k
1 vote
1 answer
92 views

A technical difficulty in the proof of Clifford’s theorem in Stichtenoth’s book

In Stichtenoth’s book (Algeraic function fields and codes) I have encountered a small technical difficulty in a Lemma ($\textbf{Lemma 1.6.14}$) that proves the following: Given a function field $F/K$ ...
Alonso Babuhicrik's user avatar
1 vote
0 answers
27 views

Zeroes of functions in elliptic function fields

Let $K/\mathbb Q$ be the function fiels defined by the affine curve $y^2=D(x)$ with $D(x)=x(x-1)(x-2)$. I want to prove that for any irreducible polynomial $Q\in\mathbb Q[X]$ inert in $K$, there is no ...
joaopa's user avatar
  • 1,149
2 votes
2 answers
171 views

what is the meaning of the notation $f: A\times B \rightarrow C$

I'm reading the 4th paragraph on page 8, under section 1.2 Fields, in the following book https://drive.google.com/file/d/1KQ7dbLXI4x39VwZovTL0DKRsZwt_i3Vt/edit "linear algebra done openly". ...
lahiru lowe's user avatar
2 votes
1 answer
153 views

Magma - Coercing a Function Field element to a Rational Function

Here's an example of what I'm trying to do: I have an elliptic curve $E : y^2 = x^3+x$ over the field $F= GF(43)$. I want to be able to go back and forth between $F(E)$ and $F(x,y)$ in MAGMA. This is ...
Rdrr's user avatar
  • 896
1 vote
0 answers
74 views

Least Prime Ideal in Chebotarev density theorem, in function fields.

For every finite extension $K$ of $\mathbb{Q}$, for every finite Galois extension $L$ of $K$ and every conjugacy class $\mathcal{C}$ of Gal$(L/K)$, there exists a prime ideal $\mathfrak{p}$ of $K$ ...
HumbleStudent's user avatar
1 vote
1 answer
83 views

Commuting polynomials in twisted polynomial ring with constant terms satisfying a polynomial relation

Suppose $q$ is a prime power, and let $A=\mathbb{F}_q[x,y]/f(x,y)$ where $f(x,y)$ is an irreducible polynomial. Let $K$ be any field such that $A$ injectively maps into $K$. (For ease of notation, ...
confused_wallet's user avatar
0 votes
0 answers
123 views

definition of an adele of a function field

From the book "Algebraic function fields and codes" of Stichtnoth, we have the definition that i can't seem to understand : where $\mathbb{P}_F$ is the set of places of the function field $...
Donnie Darko's user avatar
2 votes
1 answer
87 views

Computing cusps of the Drinfeld modular curve $X_1(t^2)$

Let $K$ be the global function field $\mathbb{F}_3(t)$ and set $$ \Gamma_1(t^2)=\left\{\begin{pmatrix} a & b\\ c & d \end{pmatrix}\in \text{GL}_2(\mathbb{F}_3[t]) : a\equiv 1 \pmod{t^2} \text{ ...
Tristan Phillips's user avatar
1 vote
1 answer
273 views

Are Weil differentials on a Function Field same as the differentials on the corresponding Riemann surface?

I was reading "Number Theory on Function Fields" by Michael Rosen, and there is a notion of "Weil differentials" on a Function Field. It intuitively seems to me that a function ...
Yashi Jain's user avatar
1 vote
0 answers
21 views

If $f(t,x)\in F_q(t)[x]$ is a Morse function, does this mean splitting field of $f(t,x)$ over $F_q(t)$ is a regular extension?

One of the classical result of Hilbert says, if $f$ is a Morse function, then the splitting field of $f(X,T)$ over $Q(T)$ is a regular extension with Galois group $S_n.$ J. P.Serre- Topics in Galois ...
Math123's user avatar
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