# 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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### Existence of solutions of polynomials systems (and their “rough” shape) over $\mathbb{R}$ & friends with positive-dimensional ideals

This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...
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### How to find a finite splitting field $K$ for $G$ such that every indecomposable $KG$-module is absolutely indecomposable

Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$. If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending ...
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### Can nonstandard fields contain $\mathbb R$ in different ways?

Suppose $e : \mathbb R \to F$ is an elementary embedding in the language of ordered fields. Can there exist an elementary embedding $e' : \mathbb R \to F$ such that $e \not= e'$? Note that it would ...
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### Isomorphism of Archimedean real closed fields with infinite transcendence degree

Are all countable Archimedean real closed fields with infinite transcendence degree isomorphic?
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### Partitioning a given set into root set of two polynomials or non zero set of two polynomials simultaneously

Let $p,q$ be two prime numbers and let $p<q.$ Also let, $\mathbb{Z}_p$ and $\mathbb{Z}_q$ denote the fields formed by integers modulo $p$ and modulo $q$ respectively (with respect to the modulo $p$ ...
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### Field extensions in Grothendieck rings

Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes ...
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### Extensions of fraction field and residue field

Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$? This question should be easy ...
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### Tensor product of fields 2

Let $K_1, K_2$ be finite field extensions of a field $k$. Question: Is it true that $A=K_1 \otimes_k K_2$ is isomorphic to a product of group algebras over fields? Question 2: In case the ...
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### Fields that are not finite extensions of proper subfields

What fields are not finite extensions of proper subfields? Prime fields and (less obviously) real closed fields have this property. Do the $p$-adics enjoy this property as well?
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### Certain $u,v,w \in \mathbb{C}[x,y]$ such that $\mathbb{C}(u,v,w)=\mathbb{C}(x,y)$

Let $\beta$ be the following involution on $\mathbb{C}[x,y]$, $\beta: (x,y) \mapsto (x,-y)$. Assume that $s_1,s_2 \in S_{\beta}$ and $k \in K_{\beta}$ satisfy: (i) $s_1,s_2$ are algebraically ...
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### Definability of the ring of integer in algebraic extensions of $\mathbb Q$

J. Robinson has proved that exists a formula $\psi(x)$ in the language of rings which,applied to the rational numbers, defines the the ring integers (making the theory of $\mathbb{Q}$ undecidable, due ...
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### Fundamental group of the Grothendieck ring scheme

Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties with $k$ a field. Let S$_k$ be its affine scheme. Is this a connected scheme for any field ? (I understand that this could be a very naive ...
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### Ultrapower of a field is purely transcendental

Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$? According to Chapter VII, Exercise 3.6 from Barnes, Mack "...
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### How many algebraic closures can a field have?

Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...