# Tagged 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $E$?
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### 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 ...
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 ...
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 ...