# 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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### Galois action on blow-ups related to field extensions of infinite degree

Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...

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### On reducing degree-$12$ equations with Mathieu group $M_{11}$ to its degree-$11$ resolvent?

The general $4$-deg and some $8$-deg (such as the Schein octic) when a linear transformation is done so their $x^{n-1}$ term vanishes can have a neat solution as,
$$x = \sqrt{z_1}+\sqrt{z_2}+\sqrt{z_3}...

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### The map from the decomposition group to the Galois group of the residue fields

$\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a Galois field extension. Let $L$ be a field and $K$ be a number field. Let $B$ be a valuation subring of $L$ and let $A$ be the preimage of $B$ in $K$ (i.e ...

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### Automorphisms of algebraically closed fields without the Axiom of Choice

In the paper Algebraische Konsequenzen des Determiniertheits-Axioms (U. Felgner – K. Schulz, Arch. Math. (Basel) 42 (1984), pp. 557–563), the authors show that in models of Zermelo-Fraenkel set theory ...

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### Automorphisms of vector spaces and the complex numbers without choice

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:
there are vector spaces without a basis;
the field of complex numbers $\mathbb{...

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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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### Understanding the inequivalent valuations inducing same topology on a field

(Crossposted from M.SE)
Few definitions:
Let $F$ be a field. We define a valuation on $F$ as a function $v:K\to G\cup \infty$ (where $(G,+,<)$ is an ordered abelian group) that satisfy:
$v(a)=\...

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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 ...

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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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### 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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### Can one $n$-sect a general angle using a "rods and hinges" construction?

This question was inspired by a post by Alon Amit.
It is a standard algebra result that it is impossible to $n$-sect a given angle using only a ruler and compass. In fact, it is impossible to trisect ...

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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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### A simple proof of the fundamental theorem of Galois theory

Update. It's now on the arXiv.
Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite ...

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### Question concerning relationships between different $p$-modular systems and Brauer character values

Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...

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### PAC and totally real fields

A field $K$ is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety over $K$ has a $K$-point. Let $L$ be the maximal totally real subfield of $\overline{\mathbb Q}$. A few ...

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### The map $k \mapsto \mathbf{PGL}_2(k)$

Consider the map $\zeta: \{ \mbox{division rings} \} \mapsto \{ \mbox{groups} \}: k \mapsto \mathbf{PGL}_2(k)$.
Is this map known to be an injection - in other words, if $k$ and $k'$ are nonisomorphic ...

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### Minor Lefschetz principle

I once read (I think) the following equivalent formulation of the Minor Lefschetz principle:
If an elementary sentence holds for one algebraically closed field,
then it holds for every algebraically ...

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### Fields with restrictions on their finite extensions: Given $n\in\mathbb{N}_{>1}$ which fields $F$ do not have extensions of degree $n$?

$\DeclareMathOperator\char{char}$This question is inspired by the MSE question Example of a non-algebraically closed field without quadratic extensions. To repeat:
Given $n\in\mathbb{N}_{>1}$ ...

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### Reference for countable and uncountable algebraic closures of $\mathbb{Q}$ in ZF

The following facts seem to be part of the folklore (where $\mathsf{ZF}$ means Zermelo-Fraenkel set theory with no axiom of choice):
it is consistent with $\mathsf{ZF}$ that there exists an ...

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### Example of a bounded imperfect PAC field that is not separably closed

How do you construct a bounded (meaning there are only finitely many separable finite extensions of any given degree) imperfect pseudo-algebraically closed field that is not separably closed? I assume ...

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### Does the choice of the algebraically closed field of characteristic $p$ have influence on the module category?

Let $G$ be a finite group and $p$ be a prime number dividing $|G|$.
Let $k$ be the algebraic closure of $\mathbb{F}_p$.
Let $K$ be another algebraically closed field of characteristic $p$ which is not ...

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### Is the group $\mathrm{Gal}(\mathbb{C}/\overline{\mathbb{Q}})$ known?

The automorphism group of the complex numbers $\mathbb{C}$ and the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are amongst the most mysterious and worst understood objects in Galois ...

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### Completion of infinite degree extension of perfectoid fields is perfectoid?

Is completion of infinite degree extension of perfectoid fields perfectoid ?
It is known that finite extension of perfectoid fields is also perftoid from tilting correspondence, but what about ...

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### Embeddings and images of number fields in $\mathbb{C}$ [closed]

Let $\mathbb{Q}(\alpha)$ be a number field, and suppose that $[\mathbb{Q}(\alpha) : \mathbb{Q} ] = m$. Then there are precisely $m$ different embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$, ...

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### Given a partition of a field, construct a partition of its extension

The motivation for my question is the following algebraic consequence of the Continuum Hypothesis ($2^{\aleph_0}=\aleph_1$) by Zoli:
(T1) Assume the Continuum Hypothesis holds. Then $\mathbb{R}^{\...

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### How hard must "no high-degree irreducibles" proofs be?

Let $\mathsf{RCF}$ be the usual theory of real closed fields and for $n>2$ let $\theta_n$ be the sentence "No degree-$n$ polynomial is irreducible." Since $\mathsf{RCF}$ is complete, for ...

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### Field extensions over which algebraic varieties cannot acquire points

The following fact (slightly reworded here) is proven in this answer:
If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite type over $...

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### A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$.
Recursive definition of addition:
$$x \oplus y := ((x+y) \...

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### What's the number of irreducible polynomials over rational numbers with constraint on coefficients?

Suppose you have polynomials $f(x) = \sum\limits_{k=0}^n a_k x^k$ where $f \in \mathbb{Q}[x]$ and $a_k$ can be either $0$ or $1$. There are $2^n$ such polynomials in total. How can I find a number of ...

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### Factorization of an irreducible polynomial in the field extension it defines

In field theory, the following fact is used in the construction of splitting fields: Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field ...

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### How to determine the degree of a rational function field over a relatively algebraic subfield?

Let $K$ be a field and $K(x_1,\cdots,x_n)$ be the degree-$n$ purely transcendental extension of $K$. Given homogeneous polynomials $f_1,\cdots,f_n\in K[x_1,\cdots,x_n]\setminus K$ with $\deg f_i=d_i$,...

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### Why is it so hard to give examples of differentially closed fields?

The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...

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### What direction does the derivation of an inseparable algebraic variable point in?

I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...

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### In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?

In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?
Particularly, is there an element $w$ of the field such that the ...

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### Indeterminacy locus of an algebraic function

Let $K=\mathbb{C}(t_1,\dots,t_n)$ be the field of rational functions, $f$ an algebraic function over $K$ and assume the field extension $K(f)/K$ is non-solvable. Is it possible to characterise the ...

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### Common Galois extension over $\mathbb Q $

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...

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### A variation on Abhyankar–Moh–Suzuki theorem

The well-known theorem of Abhyankar–Moh–Suzuki says the following:
Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
If $k[f,g]=k[t]$, then $\deg(f) \mid \deg(g)$ or $\deg(g) \mid \...

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### Proving that polynomials belonging to a certain family are reducible

In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone.
Assume that $\mathbb F_p$ is ...

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### Lines on quadric surfaces

Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?

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### Abelian approximation of fields

Given a field extension $K/k$ of finite degree and a norm $d$ on $\overline{k}$, what is the smallest real number $\alpha_{d}^K$ such that for every element $z$ of $K$ there is an element $z^a$ of $k^...

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### A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...

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### Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?

Consider the hierarchy of relative geometric constructibility by
straightedge and compass. Namely, given a geometric figure $B$, a
set of points in the plane, we define that geometric figure $A$ is
...

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### Algebraic topology over fields other than ${\bf R}$

Is there an algebraic topology for spaces defined
on fields other than ${\bf R}$, including totally discontinuous fields ?
By this, I am not talking about the field of coefficients,
but about the ...

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### If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?

I posed this question on Math.Stackexchange (see here) but until now there was no response. This made me decide to give it a try here.
Let $k\subseteq F$ denote an algebraic field extension and let $\...

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### Topological field isomorphic to $\mathbb{C}$

Let $K$ be a topological field. If $K$ is a connected Hausdorff space, and is algebraically closed, is it true that $K$ is isomorphic to $\mathbb{C}$ ?
(I have deleted my question on MathStackExchange)...

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### Are there only two smooth manifolds with field structure: real numbers and complex numbers?

Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - $\mathbb{R}$ ...

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### Existence of an irreducible polynomial that does not divide $x^n + a$

The question:
Can one characterize all fields $K$ over which there exists an irreducible polynomial $f(x)$ that does not divide a polynomial of the form $x^n + a$?
Examples:
Such a polynomial clearly ...

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### Results in linear algebra that depend on the choice of field

Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers).
I am looking for a list of concepts, and results, in ...