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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 ...
THC's user avatar
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-3 votes
1 answer
162 views

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 $R$ ...
user237522's user avatar
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1 vote
0 answers
96 views

A certain subfield of $\mathbb{C}(x,y)$

Let $A=\mathbb{C}(x+y,xy)$, the subfield of symmetric polynomials with respect to the involution $\alpha: (x,y) \mapsto (y,x)$. Denote $G_A=\{w \in \mathbb{C}(x,y) \ | \ \mathbb{C}(x+y,xy,w)=A(w)=\...
user237522's user avatar
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5 votes
1 answer
297 views

Surjection onto endomorphisms of multiplicative group of a field

Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$ $$ \mathbb{...
Nicholas's user avatar
5 votes
1 answer
154 views

Can one compute the automorphism group of a curve of genus >1?

Given a sufficiently nice perfect field $k$ and a smooth projective curve $C$ of genus $g_C>1$ over $k$, can one compute the automorphism group ${\rm Aut}(C)$? It is known that ${\rm Aut}(C)$ is ...
Arno Fehm's user avatar
  • 2,041
5 votes
2 answers
677 views

A version of Hilbert's Nullstellensatz for real zeros

$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
Iosif Pinelis's user avatar
2 votes
0 answers
75 views

The involutive structure on a division ring

This question is motivated by foundations of geometry, namely, by studying scalars in affine spaces. Let $F$ be a field (or better a division ring). It has the operations of addition and ...
Taras Banakh's user avatar
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21 votes
1 answer
1k views

Are the real numbers isomorphic to a nontrivial ultraproduct of fields?

Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$. Question: Can there be a field ...
Tim Campion's user avatar
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5 votes
1 answer
244 views

Converse of "generalized Hilbert 90" / Galois descent

The following generalization of Hilbert 90 can be found in Serre's Corps Locaux (Chap. X, §1, ex.2, p.160 of the French edition), see also this question: Theorem: If $L|K$ is a finite Galois extension ...
Béranger Seguin's user avatar
3 votes
0 answers
329 views

Semidirect product in inverse Galois problem

Let $L/\mathbb{Q}$ (resp. $K/\mathbb{Q}$) be a Galois extension of rational number field $\mathbb{Q}$ with Galois group $P$ (resp. $H$) where $P$ is a second countable pro-$p$ group and $H$ is a ...
stupid boy's user avatar
2 votes
1 answer
93 views

Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$

Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$. I am looking for a simple proof of the following fact. "If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
Oblomov's user avatar
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0 answers
104 views

Generalization of SVD algorithm

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
Martin Clever's user avatar
0 votes
0 answers
103 views

Can a definable group of definable automorphisms of a field contain the Frobenius automorphism?

Let $K$ be an infinite definable field of characteristic $p >0$ in a certain theory $T$ with a definable group of definable automorphisms. Can this group contain the Frobenius automorphism?
Invictus's user avatar
6 votes
1 answer
158 views

Constructing countable threelds of finite dimension

A threeld is a generalization of a field, with three operations, such that the $F$ is a field with respect to the first (outer) and second (middle) operations (call it the outer field), and $F\...
Thomas's user avatar
  • 2,751
5 votes
2 answers
467 views

When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?

When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension? For example, does $[\mathbb{Q}(\sqrt[n]{2},\sqrt[m]{3}):\mathbb{Q}]=mn$ hold true? Are there more ...
sofia's user avatar
  • 51
1 vote
0 answers
110 views

Select random point on elliptic curve

If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...
R Artur's user avatar
  • 11
6 votes
0 answers
434 views

Extensions of p-adic number fields

Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...
Eric's user avatar
  • 71
3 votes
0 answers
71 views

Locally compact rings with reciprocals

A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
Andre Kornell's user avatar
5 votes
2 answers
359 views

Can every finitely generated field extension of $\mathbb{Q}$ be embedded into a local field?

Let $K$ be a finitely generated field extension of $\mathbb{Q}$, and let $p$ be a prime number. Can $K$ must be embedded into a p-adic local field (i.e. a finite field extension of $\mathbb{Q}_p$) ?
lolipop's user avatar
  • 53
5 votes
1 answer
475 views

General algebraic result obtained from consideration on $\mathbb{Q}_p$

There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields. For instance, the fact that a polynomial $P$ admits a ...
Weier's user avatar
  • 151
8 votes
1 answer
191 views

Sequential colimit of iterated quotients of Cauchy sequences

We work in constructive mathematics. The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
Madeleine Birchfield's user avatar
4 votes
1 answer
195 views

Subfields of division rings of degree $2$ which are not invariant

Let $A$ be a noncommutative division ring, and let $B$ be a sub division ring (here, $B$ is allowed to be commutative) of degree $2$. Are there easy examples known for which $B$ is not globally fixed ...
THC's user avatar
  • 4,423
-1 votes
1 answer
292 views

Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?

The idea for the following question came from Joachim König's last comment appearing here, namely, the example with $u=x+y^3,v=x^3+y$. Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
user237522's user avatar
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0 votes
1 answer
134 views

$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$. Claim: $\mathbb{C}(...
user237522's user avatar
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0 votes
1 answer
143 views

$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?

The following question is a direct continuation of this question: Let $u,v \in \mathbb{C}[x,y]$. Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
user237522's user avatar
  • 2,789
4 votes
1 answer
243 views

If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$?

The following question is a direct continuation of this elaborate question; it is mentioned there at the end: Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in ...
user237522's user avatar
  • 2,789
2 votes
1 answer
212 views

Chinese remainder theorem for composition

Let $f(x) \in F_p[x]$ and I know $f(x)$ modulo two polynomials $\phi_1(x)$ and $\phi_2(x)$. What sort of information about $f$ modulo the composition $\phi_1(\phi_2(x))$ can I recover? I'm looking ...
mtheorylord's user avatar
6 votes
1 answer
239 views

If degree $N$ polynomials always have a root, does there still exist an irreducible polynomial of degree $N+1$?

Given that $F$ is a field, let $F_n$ be the completion of $F$ with respect to roots of degree $n$ polynomials. For example this would make $\mathbb{Q}_2$ the field of (ruler and compass) constructible ...
Sam Forster's user avatar
4 votes
1 answer
213 views

Finiteness of number of extensions with bounded degree and discriminant

Fix natural numbers $d,N$ and a polynomial $\Delta \in \mathbb{C}[x_1,\ldots,x_d]$. Let $S_{d,N}$ be the set of field extensions $K/ \mathbb{C}(x_1,\ldots,x_d)$ such that The degree $[K: \mathbb{C}(...
Rami's user avatar
  • 2,591
3 votes
1 answer
956 views

irreducibility of the polynomial $ x^4 +1 $

Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
Sky's user avatar
  • 913
1 vote
0 answers
216 views

Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero

When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
Béla Fürdőház 's user avatar
2 votes
1 answer
393 views

Algebraically closed fields with only finite orbits

The automorphism groups $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ of algebraic closures of finite fields $\mathbb{F}_q$ and the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\...
THC's user avatar
  • 4,423
1 vote
0 answers
122 views

Interplay between additive and multiplicative characters of fields

Let $F$ be a countable field. Given a double Folner sequence $(F_N)_{N\in \mathbb{N}}$ in $F$, an additive character $\chi$ (i.e. $\chi: F \to \mathbb{S}_1$ satisfies $\chi(u+v)=\chi(u)\chi(v)$, for ...
User's user avatar
  • 153
0 votes
1 answer
144 views

Not containing $ 2^{i}$-th primitive roots of unity in cyclic galois extension of number field

Suppose $K$ is an algebraic number field. We want to determine whether, for all even $n$, there always exists a cyclic Galois extension $L$ of degree $n$ over $K$ such that the intermediate field $K_2$...
Sky's user avatar
  • 913
0 votes
0 answers
44 views

Transcendence of functions and change of field of definition

Suppose the one has a sequence of rational functions $Q_n(z)\in\mathbb Q(z)$. Let $p$ be a prime number. Suppose that that there exists an infinite subset $X$ of $\mathbb Q_p$ such that: the ...
joaopa's user avatar
  • 3,801
2 votes
0 answers
122 views

Can global fields be defined as certain topological fields like local fields?

It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...
Z Wu's user avatar
  • 340
6 votes
0 answers
146 views

Proof of Theorem Concerning Conway's "Nim Field"

I have a question about the proof of theorem 44 in "On Numbers and Games" on page 58, concerning the "Nim field" ${ON}_2$. As background, ${ON}_2$ turns the ordinals into a field ...
interstice's user avatar
1 vote
0 answers
52 views

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 $\mathrm{Gal}(\ell/k)$ ...
THC's user avatar
  • 4,423
1 vote
1 answer
97 views

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}...
Tito Piezas III's user avatar
1 vote
0 answers
154 views

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 ...
M. K.'s user avatar
  • 45
6 votes
1 answer
481 views

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 ...
THC's user avatar
  • 4,423
4 votes
1 answer
322 views

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{...
THC's user avatar
  • 4,423
11 votes
0 answers
280 views

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 $g(x)$ ...
Noah Schweber's user avatar
0 votes
0 answers
27 views

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 ...
Menander I's user avatar
1 vote
0 answers
93 views

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 ...
Ali Taghavi's user avatar
-2 votes
1 answer
147 views

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 ...
Yoav Len's user avatar
  • 147
5 votes
1 answer
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 ...
Tito Piezas III's user avatar
4 votes
3 answers
467 views

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; ...
Madeleine Birchfield's user avatar
6 votes
2 answers
211 views

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 ...
Boaz Moerman's user avatar
27 votes
1 answer
2k views

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 ...
Martin Brandenburg's user avatar

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