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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.

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Primes Splitting in the Gaussian Integers - Function Field Analogue

The function field analogue of the ring $\mathbb{Z}[i]$ are functions of the form $A(T)+\alpha B(T)$, where $\alpha$ is a solution to the equation $x^{2}+T=0$, over $F_{q}[T]$. We know that a prime $p$ splits over $\mathbb{Z}[i]$ if and only if $p…
Ezra
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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 would have that $L=K(g(x),h(x))=K(x)$. This does not happen…
Reyx_0
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genus of the field $\mathbb{R}(x,y)$ where $x^2+y^2+1=0$

I have come across a problem while reading Chevelley's "Introduction to the theory of algebraic functions of one variable", in which he says that the genus of the field $L=\mathbb{R}(x,y)$ is 0, where $x$ is transcendental over $\mathbb{R}$ and $y$…
bharath
  • 169