The function-fields tag has no wiki summary.
9
votes
1answer
172 views
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 ...
8
votes
1answer
154 views
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 ...
4
votes
2answers
136 views
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 ...
4
votes
1answer
55 views
Levels of Rings and Fields, -1 as a sum of squares
Definition: Let $R$ be a commutative ring. The level of $R$, denoted $s(R)$, is the least positive integer $s$ such that $-1$ can be written as the sum of $s$ many squares in $R$. Set $s(R)=\infty$ if ...
3
votes
1answer
72 views
what is the constant field of irreducible components a divisor?
Let $D$ be a divisor on an algebraic variety over a field $k$, that is
$$
D=\sum n_i D_i
$$ where $D_i$ are the irreducible components. I came across the expression "the constant field of $D_i$" and ...
3
votes
1answer
94 views
A field isomorphism related to polynomial rings and their field of fractions
There are 2 ways to approach function fields: the algebraic approach, i.e. looking at finite extensions of $K(s)$, where $s$ is transcendental. The other is geometric, i.e. considering functions over ...
3
votes
1answer
112 views
On Intermediate Fields of $\mathbb{C}(x_1,\dots,x_n)$
I am recently reading some Galois Theory, and a question occurred to me: What are the intermediate fields of $K$ of $\mathbb C(x_1,\dots,x_n)$, where $n$ is an arbitrary integer?
I am aware of a ...
2
votes
1answer
73 views
Why do number rings have no endomorphisms
This question is about the analogy between number fields and function fields. It's a soft question and the title misrepresents the question.
Consider the projective line over a field. This has many ...
2
votes
1answer
124 views
Completion of a rational function field w.r.t. place at infinity
When taking the completion a rational function field, say $k(t)$, with respect to the place at infinity, most books refer to this using the notation $k((1/t))$.
Since $k((t)) = k((1/t))$ (EDIT: this ...
2
votes
1answer
124 views
Two notions of uniformizer
Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
2
votes
1answer
144 views
Hyper-elliptic curves in positive characteristic
I have been looking at hyperelliptic curves in characteristic two, in particular using Algebraic Geometry and Arithmetic Curves by Qing Liu, which gives a description in all characteristics.
For the ...
2
votes
2answers
181 views
Existence of morphism of curves such that field extension degree > any possible ramification?
Throughout I would like to work over an algebraically closed field of characteristic 0 (so no separability issues), say $k$. My question is the following:
Do there exist two curves $X$ and $Y$ and a ...
2
votes
1answer
68 views
Linear disjointness of two “explicit” field extensions
Let $k$ be a characteristic zero field and let $L/k$ be a quadratic extension. Write $L = k(\sqrt{p})$.
Let $q$ be a non-square in $k^\star$ and let $r \in k^\star$ be any constant.
Consider the ...
2
votes
0answers
40 views
Algebraic Curves similar to Hyper-Elliptic Curves
Throughout, $F_q$ will denote a finite field of $q$ elements with characteristic $p \neq 2$.
It is well-known that the equation $y^2 = f(x)$ (for square-free $f \in F_q[X]$) defines an hyper-elliptic ...
2
votes
1answer
56 views
Residue map of a place
The following definition for a residue map is given in "Algebraic Curves over a Finite Field" by Hirschfeld, Korchmáros and Torres (page 265):
"Let $\Sigma$ be a field of transcendence degree 1 over ...
1
vote
1answer
87 views
extension of algebraic function field
Let $K$ be a field, $t$ a transcendetal element over $K$ and $F'|K(t)$ an infinite Galois extension. Hence I have a tower of extension $K\subset K(t) \subset F' $.
Does exist a subextension $F$ of ...
1
vote
1answer
122 views
Separability of compositum of fields
Let $E/F$ be a finite separable extension, and let $K$ be a function field with constant field $F$. Is the compositum $KE$ of $K$ and $E$ a separable extension over $E$?
1
vote
0answers
185 views
Artin-Schreier extensions over characteristic two fields
I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do ...
1
vote
0answers
85 views
Calculating degree of a finite Kummer Extension
Assume I have a field $K$ containing all $n$-th roots of unity. You may even assume that $K$ contains an algebraically closed field $\Bbbk$. Assume furthermore that there are $x_1,\ldots,x_k\in K$ and ...
0
votes
0answers
32 views
places of function field and closed point of a scheme
Given an integral scheme $X$, let $K(X)=\mathrm{Frac}(R)$ be its function field, where $\mathrm{Spec}(R)$ is some non-empty open affine subscheme of $X$. Take the maximal ideal $P$ of some DVR of ...
