# Tagged Questions

Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

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### question about a particular Polynomial ring [closed]

Let K be a field, let $T = K[X_1, X_2,...]$ be a polynomial ring, let $R=K[X_1^{2}, X_1X_2,..,X_i X_j,..]$, and let $L = Frac(R)$ = field of fractions of R. How can we prove that $R =T \cap L$ ?
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### Is there a symmetric basis for $\mathbf{Q}(x,y)$?

Consider $\mathbf{Q}(x,y)$, the rational functions in $x$ and $y$, as a vector space over $\mathbf{Q}$. Let $\sigma$ be the map interchanging $x$ and $y$. Is there a basis for $\mathbf{Q}(x,y)$ ...
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### On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$

(Major revision to incorporate new results in this MO cubic version.) Note: All coefficients are in the rationals. I. Cubic In the linked post, it was shown that given a general cubic (via its ...
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### Does k(X) have a k-basis for every set X, without AC?

This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?. For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
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### Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
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### Variety with perfect function field?

My question is quite simple: Let $X$ be an irreducible algebraic variety over a field $\Bbbk$. Is there a name for such varieties with perfect function field $\Bbbk(X)$? Is this very rare? Is there ...
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### Rationality of curve does not depend on base change

By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field. Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...
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### On the solvable octic $x^8-x^7+29x^2+29=0$
The irreducible but solvable octic, $$x^8-x^7+29x^2+29=0\tag{1}$$ was first mentioned by Igor Schein in this 1999 sci.math post. This does not factor over a quadratic or quartic extension, but over ...