# Questions tagged [fields]

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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### Squares in skew fields of dimension 2 over a sub skew field

Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$. Then if $a \in \ell \setminus k$, we can ...
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### A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$

Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$. Here $\mathbb{N}$ includes $0$. Assume that $R$ ...
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### Can every "not-too-big" differential field be thought of as actually consisting of functions?

Previously asked and bountied at MSE without success: Let $\sim$ denote the "no-disagreement" relation between partial functions: $f\sim g$ iff there is no $x$ such that $f(x)$ and $g(x)$ ...
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### Alternative equivalence results for the constructibility of real numbers

Everyone is aware of the standard result from undergraduate field theory that a real number $\alpha$ is constructible by straightedge and compass if and only if there exists a finite sequence of field ...
• 273
1 vote
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### The asymptotic growth of codimension of range of polynomial differential equation on finite fields

Inspired by the seminal paper of Andre Weil on the number of solutions of equations on finite fields we would like to present the following question: Let $P(x,y), Q(x,y)$ be two polynomials of ...
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### Finite normal extensions

Suppose that $K$ is a finite field extension of $F$. Is the following equivalent to the extension being normal? If $L$ is an extension of $K$ and $\sigma:K\to L$ fixes $F$, then $\sigma(K) = K$. I ...
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273 views

### Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?

I. Kondo-Brumer quintic The deceptively simple solvable quintic, $$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$ is quite important for imaginary quadratic fields. For ...
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### The field structure on the locale of real numbers

It is well known how to derive the field operations from the construction of the real numbers as the Dedekind completion of the rational numbers and as the Cauchy completion of the rational numbers; ...
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### For which fields $k$ with $k^\times$ not $p$-divisible, does there exist finite $l/k$ such that $l^\times$ is $p$-divisible?

Is there a prime $p$ and a field $k$, not real closed, with $k^\times$ not $p$-divisible, such that there exists a finite extension $l/k$ such that $l^\times$ is $p$-divisible? This question came up ...
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