Questions tagged [function-fields]

This tag is for questions related to function field, a finitely generated field extension.

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Meaning of absolutely additive

What is the meaning of the polynomial is 'absolutely additive?' More precisely, let $k$ be a field with characteristic $p$ and $\overline{k}$ be its algebraic closure. Goss' book 'basic structure of ...
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22 views

A proof of existence of canonical divisors

I am confused with the proof of Lemma 1.5.10 in Algebraic Function Fields and Codes by Henning Stichtenoth. Let $0\ne\omega\in\Omega_F$. Then there is a uniquely determined divisor $W\in M(\omega)$ ...
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1answer
45 views

Residue fields at points on $\mathbb{A}^n$

Let $k=\bar k$ be a field. I'm trying to "write down" the residue fields at various points on $\mathbb{A}^n=\operatorname{Spec} k[x_1,\cdots, x_n]$, but am having some trouble with the non-...
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38 views

Solubility of Cubic Equation over $\mathbb{F}_q[t]$

Hello fellow mathematicians, I have the following problem: Let $u_1X^3 + u_2Y^3 + u_3Z^3 - cXYZ = 0$ be a homogeneous cubic equation, where the coefficients $u_1$, $u_2$, $u_3$ and $c$ are in $\...
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1answer
43 views

Genus $3$ curves with a couple of distinct points $P,Q$ such that $4P \sim 4Q$

Let $C$ be a smooth curve of genus $3$ over $\mathbb{C}$. Is it true that there exist $P\neq Q \in C$ such that $4P \sim 4Q$ ? ($\sim$ denotes linear equivalence) Notice that if $C$ is hyperelliptic ...
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86 views

Function field analogy

I am rephrasing the previous question: Can I get good and accessible references to read to understand this particular statement in Wikipedia: "The function field analogy states that almost all ...
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10 views

Are there upper bounds for the genus of an extension of function fields, can you have unbounded genus growth?

If $F$ is a function field with constant field $K$, and $E$ is a finite extensions of $F$, then Riemann-Hurwitz gives a way to compute the genus, $g_E$, of $E$ from the genus, $g_F$, of $F$ so long as ...
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53 views

Is there any easy proof for this “weak version” of Mazur's theorem on the torsion group of $\mathbb{Q}$-elliptic curves?

I am dealing with the following problem: Given a $\mathbb{Q}\left (t\right )$-elliptic curve $E(t)$ with positive rank, prove that for all but finitely many $t_0\in \mathbb{Q}$, the $\mathbb{Q}$-...
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47 views

Understanding the group of divisors of a function field and when two extensions of function fields isomorphic

I am reading Stichtenoth's book on function fields and codes and have just arrived at the discussion of divisors and the divisor class groups for function fields. I'm interested in how these are in ...
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11 views

Algebraic closure of completions in function fields

Consider $k= \mathbb{F}_q(t)$ and let $p$ be an irreducible polynomial in $\mathbb{F}_q[t]$. Consider the completions of $k$ with respect to $p$ and the $\infty$-adic completion, with notation $k_p$ ...
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What is the number/structure of units in a function field over $\mathbb{F}_q$?

I know that the units of $\mathbb{F}_q[T] \subset \mathbb{F}_q(T)$ are the constant non-zero polynomials, of which are there $q-1$, and you can think of these as being in analogy with the units $\pm1$ ...
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1answer
45 views

Books for function fields and algebraic curves

I'm just finished my learning of algebraic number theory, and I want to go for some algebraic geometry which I basically know nothing about. At the same time, I'm informed that there're lots of ...
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33 views

Imaginary quadratic extensions in characteristic 2

Let $\mathbb{F}_q$ be finite field with characteristic 2 and consider the polynomial ring $A=\mathbb{F}_q[T]$. Denote $F=\mathbb{F}_q(T)$. Let $f(x)=ax^2+bx+c\in A[x]$ be irreducible. I'm trying to ...
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1answer
55 views

transcendental extension $\mathbb{C}(x)$ over $\mathbb{C}$, finding fixed field

Let $K=\mathbb{C}(x)$ where $x$ is transcendental over $\mathbb{C}$. Let $\sigma$ be the automorphism of $K$ over $\mathbb{C}$ given by $\sigma(x)=\zeta x$ where $\zeta$ is a primitive 3rd root of ...
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1answer
83 views

$f(t,x)$ irreducible, unsolvable over $\mathbb Q(t)$ such that its specialization $f(t_0,t)$ over $\mathbb Q$ is inseparable and (un)solvable.

The following theorem can be found in e.g. Lang or Van Der Waerden. Let $f(t,x)\in\mathbb Q[t,x]$ be irreducible with Galois group $G$ over $\mathbb Q(t)$. If $t\to t_0\in\mathbb Q$ is a ...
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1answer
35 views

Checking the Hasse-Weil bound for genus zero

According to Stichtenoth's Algebraic Function Fields and Codes, the Hasse-Weil bound theorem is: The number $N$ of places of $F/\mathbb{F}_q$ of degree one satisfies $|N-(q+1)|\leq 2gq^{1/2}$. [here $...
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22 views

$F/\mathbb{C}(x)$ ramifies at $2$ places $\Rightarrow F=\mathbb{C}(y)$ with $y^n=\frac{ax+b}{cx+d}$

Let $F/\mathbb{C}$ be a function field such that $z\in F$ is integral over $\mathbb{C}\Leftrightarrow z\in\mathbb{C}$. Suppose there is $x\in F\setminus\mathbb{C}$ such that $\mathbb{C}(x)/\mathbb{C}$ ...
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21 views

$\mathcal{O}_S$ holomorphy ring, $S$ containing almost all places $\Rightarrow O_S=K[x_1,…,x_r]$

This is exercise $3.5$ from Stichtenoth's Algebraic Function Fields and Codes: Let $F/K$ be a function field in one variable and $\mathbb{P}_F$ the set of its places. If $S\subsetneq \mathbb{P}_F$ is ...
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1answer
52 views

For every function field $F/K$ there exists $x\in F$ with $F/K(x)$ separable

This is exercise $3.2$ from Stichtenoth's Algebraic Function Fields and Codes: Let $F/K$ be function field in one variable such that $K$ is perfect and $\overline{K}\cap F=K$. Given $P_1,...,P_r$ ...
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30 views

If $F'/K'$ is an algebraic extension of $F/K$, then $K'/K$ is algebraic

Let $F'/K'$ and $F/K$ be function fields in one variable such that $K,K'$ are the full constant fields of $F,F'$ respectively. I'm trying to understand the following statement from lemma $3.1.2$ in ...
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29 views

Exact sequence of Riemann-Roch spaces and adeles

I am studying Riemann-Roch theorem on the book by Stitchtenoth, "Algebraic function fields and codes". I am having some troubles understanding a exact sequence of linear mappings between Riemann-Roch ...
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1answer
57 views

Definition of a valuation on the points of an elliptic curves

I have to give a presentation about function fields and valuation. I will discuss the case of rational function on the projective line. Then I would like to give an example of a different function ...
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1answer
110 views

Infinite places of a global function field

Let $K$ be a global function field, that is, a finite extension of $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is the finite field with $q$ elements. What are the infinite places of $K$? I know that $...
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1answer
27 views

Function field $F/K$, then $\exists$ a divisor $A$ with $\deg(A)=g$ and $\ell(A)=1$

This is exercise 1.15 from Stichtenoth's Algebraic Function Fields and Codes: Assume that the constant field $K$ is algebraically closed. Show that for every integer $d\geq g$, there exists a ...
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29 views

If $F=\mathbb{R}(x,y)$ with $x^2+y^2+1=0$, then all places of $F/\mathbb{R}$ has degree $2$

Consider the function field $F/\mathbb{R}$, where $F:=\mathbb{R}(x,y)$ with $x^2+y^2+1=0$. Show that $F/\mathbb{R}$ has genus $0$ but is not a rational function field. Furthermore, all places of $F/\...
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52 views

Is the polynomial degree (difference) a valuation?

Let $k$ be a field and $K=k(x)=\{f/g\mid f, g\in k[x], g\ne 0\}$ the field of rational functions. Let $\mathcal{O}$ be the ring $$\mathcal{O}=\{\frac{f}{g}\in K\mid \deg f\ge\deg g \}\cup\{0\}$$ Then ...
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1answer
62 views

Factorizarion criterium to decide whether $K$ is the full constant field of $F/K$

I took the following statement from an answer in math.stackexchange: If $G(x,y)$ is an irreducible polynomial in $K[x,y]$, and $F=K(x,y)$ is an algebraic extension of $K(x)$ defined by $G(x,y)=0$, ...
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35 views

Why assume $K$ is the full constant field of $F|K$?

In the beginning section $1.4$ from Stichtenoth's Algebraic Function Fields and Codes, the author says: "From here on, $F/K$ will always denote an algebraic function field of one variable such that ...
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18 views

Witt Vectors over $\mathbb{F}_q[T]$

I have seen an article where the author constructs a ring of Witt vectors over $\mathbb{F}_q[T]$ with respect to a prime polynomial $P$. That is, it constructs the map $$(\mathbb{F}_q[T]/P)^{\mathbb{N}...
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1answer
63 views

Eisenstein criteria in k(t)[X]

Let $k$ be a field of characteristic $\ne 2$ and $L$ be a finite extension of the function field $k(t)$ in which $\sqrt{t}$ and $\sqrt{1+t}$ exist. Find the minimal polynomial of $T:=\sqrt{t}+\sqrt[...
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1answer
62 views

$F|K$ with a rational place $\Rightarrow F=K(x,y)$ with $[F:K(x)]=[F:K(y)]=2g+1$

Let $F|K$ be a function field in one variable with genus $g$. If there is a place $P$ with degree one, then $\exists\,x,y\in F$ such that $F=K(x,y)$ and $[F:K(x)]=[F:K(y)]=2g+1$. I'm really lost in ...
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73 views

For positive genus, $\ell(A)=\deg(A)+1\Leftrightarrow A$ is a principal divisor

(this is exercise $1.8$ from Stichtenoth's Algebraic Function Fields and Codes) Let $F|K$ be a function field in one variable with genus $g>0$. If $A$ is a divisor with $\ell(A)>0$, show that $\...
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1answer
90 views

Where can I find this paper by Iwasawa?

I'm trying to find a copy of the 1969 article Analogies between number fields and function fields by Iwasawa. There is a similar paper by Wiles and Mazur titled Analogies between function fields and ...
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1answer
52 views

What is the genus of an unramified extension of a function field?

Suppose L/K is a degree $n$m separable, geometric extensions of function fields where both function fields have constants $\mathbb{F}_q$. Suppose also that $L$ is unramified over $K$. Let $g_L$ be the ...
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18 views

Genus behavior under base extension

Suppose that $F$ is a function field of a smooth geometrically connected curve $C$ over $\mathbb{F}_q$. So $F$ is some function field with constants $\mathbb{F}_q$. We could instead look at $C$ over ...
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26 views

In the rational function field $K(x)|K$, the place at infinity $P_\infty$ has degree $1$

Let $P_\infty$ be the place at infinity at the rational function field $K(x)|K$. I'm trying to prove that $\deg(P_\infty)=1$ (where $\deg(P_\infty):=[\mathcal{O}_\infty/P_\infty:K]$ and $\mathcal{O}_\...
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74 views

Inertia group of function field

Let $C(T)$ be a function field in the variable $T$ over an algebraically closed field $C$ of characteristic $0$. Consider L as the splitting field for the following polynomial: $F(X,T) = f(X)-T$ ...
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2answers
153 views

For every function field $L$ there is a smooth projective curve $C$ with $L\simeq k(C)$

Let $k$ be an algebraically closed field. It is a well-known result that: The category of smooth (i.e., non-singular) projective curves with dominant morphisms is equivalent to the category of ...
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1answer
43 views

Analogue of Wilson Theorem for polynomials in a finite field.

Proposition 1.9 I'm trying to understand chapter 1 of Michael Rosen's Number Theory in Function Fields, and on this particular proposition I have questions just about everything (see Proposition 1.9)....
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1answer
129 views

Equivalence between function fields and curves

Sometime ago I overheard a conversation in which someone said "studying functions fields is the same thing as studying algebraic curves". After looking it up, I've found these two results (I'...
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1answer
42 views

Are these type spaces equivalent?

If I have a real 3-vector, I can say that is of type $\mathbb{R}^3$. But equally I say that it is a map of the index which takes one of 3 values to a real number so the type is $\mathbb{Z}_3 \...
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1answer
22 views

how to split a separable algebra?

I'm trying to factor ideals in a function field (more precisely, ideals in a maximal order of a function field), and I've come across a step in the published Buchman-Lenstra algorithm which works in ...
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0answers
33 views

Definition of Algebraic Function Field

the definition of algebraic function field $\mathbf{F}$ over $\mathbf{K}$ that I know is the following: There exists an element $x \in \mathbf{F}$, transcendental over $\mathbf{K}$, such that $\...
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26 views

An isomorphism between $A/P^{e}A$ and $A/PA$ and the direct sum of a $p$-group

I'm making some notes for Number Theory in Function Fields by Rosen, and for the life of me I cant figure out why the following is true (in the notation below $A = \mathbb{F}_{q}[T]$, $P \in A$ is ...
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25 views

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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1answer
168 views

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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21 views

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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52 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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0answers
61 views

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
52 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 ...