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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 $k$.

Definition : Let $k$ be a field. An algebraic function field (often abbreviated as function field) $K$ over $k$ is a finitely generated extension over $k$ of a finite transcendence degree at least one. If the transcendence degree of $K/k$ is $r$, we say that it is a function field in $r$ variables.

Equivalently, an algebraic function field of $n$ variables over $k$ may be defined as a finite field extension of the field $K=k(x_1,\cdots,x_n)$ of rational functions in $n$ variables over $k$.

Note :

  • Such fields emerge as fields of rational functions of an $r$-dimensional variety over the field $k$. For example, when $r=1$ the variety is a curve. Hence the name.
  • Function fields of transcendence degree one over a finite field $k=\Bbb{F}_q$ are a particularly well studied class for their properties resemble those of number fields.
  • One of the distinctions between number fields and function fields over finite fields is that the latter have no smallest subfield that is itself also a function field.

For more details:

"Handbook of Algebra" by Moshe Jarden

https://en.wikipedia.org/wiki/Algebraic_function_field

http://www.math.tifr.res.in/~publ/ln/tifr18.pdf

Number Theory in Function Fields by Michael Rosen.

238 questions
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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 holds for all finite extensions of a perfect…
Ali
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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 $\phi = \left[ f_1, \ldots, f_n \right]$ for homogeneous polynomials $f_i \in…
kless135
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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 know that $char(K)=p$ and $K$ is not a perfect…
NoBinash
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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 $\mathbb{ Q}(\alpha,i)$. Now the degree of the…
youknowwho
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Number of square-free polynomials over a finite field - a combinatorial interpretation?

One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a finite field $\Bbb F_q$ is $q^n-q^{n-1}$. For instance, this is done here. The answer is simply enough to suggest a very nice combinatorial…
Asvin
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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 generated by these derivatives is finite. Here are two…
jspecter
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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 $\mathbb F_q(t)$ is indeed $\mathbb F_q$? If one computes…
Reyx_0
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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…
Soroush Salehin
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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 a prime $P_i$ for every $a_i$- its place is…
Madarb
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What are function fields?

Given a field $F$, I know that $F[X]$ is the ring of polynomials in $X$. I know that this is not a field. I have seen the notation for $F(x)$ with round brackets. Usually when we use round brackets we take the smallest field containing $F$ and $x$.…
John Doe
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Quotients of Elliptic Curves

I am fairly inexperienced with elliptic curves so there might be aspects of my question that may need better wording but let me know if there are any issues: Question: Say I have an elliptic curve over $\mathbb{F}_7$ and this curve has 12 points. I…
Tom Lewia
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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 by finite decimal expansions and some real…
Snor
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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 inseparable (and irreducible) polynomial. Let $\mathcal{P}$…
BharatRam
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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 $f(x)-t\in K(t)[x]$ has distinct roots in…
Ferra
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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): Let $k$ a function field in one variable over…
C.D.
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