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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2
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1answer
60 views

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 ...
1
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1answer
57 views

Under what conditions is the compositum of two rational cubic Kummer extensions a rational function field?

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $F = k(x)$ be the rational function field in the variable $x$. Consider two Kummer extensions $F_1 = F(\sqrt[3]{g_1})$, $F_2 = ...
8
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2answers
443 views

What is the state-of-the-art for solving polynomials systems over fields that are not algebraically closed?

I am not working in the field of algorithmic algebraic geometry - yet, for my current work, I need some results from it. More specifically, what is the state-of-the-art when it comes to solving (...
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0answers
64 views

Is $E$ finite and normal? [closed]

Let $E$ be a field and $H$ a finite subgroup of the group Aut $E$ of all automorphisms of $E.$ Let $F=$ Inv $H$ be the field of invariants of $H$. Is $E$ finite and normal over $F$? Is every element ...
1
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1answer
92 views

Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of non-Archimedean local fields where I can find proofs of the following claims about finite extensions $L/K$ of non-Archimedean local ...
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0answers
48 views

Does “tensoring” with a fixed field preserve Galois extensions of finite fields?

Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...
5
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1answer
105 views

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 ...
8
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2answers
373 views

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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0answers
38 views

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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0answers
57 views

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 $p$ ...
3
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1answer
279 views

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 ...
3
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0answers
73 views

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 ...
2
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0answers
175 views

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 ...
6
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1answer
132 views

Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?

It is a well-known fact that the Henselization of the function field $\mathbb{F}_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}_{p}(t)^{alg} \cap \mathbb{F}_{p}((t))$, so of course $\...
3
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0answers
39 views

Bounds on degrees of minimal polynomials of infinite degree algebraic extension

If $E/F$ is algebraic extension of finite degree $n$, then if $\alpha \in E$ is an element, then the degree of minial polynomial $m_\alpha$ for $\alpha$ is at most $n$. Even better, $\deg m_\alpha$ ...
2
votes
1answer
105 views

Classification of associative polynomial functions

What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers ...
3
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1answer
95 views

Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals

Suppose $K=\mathbb{Q}(X_1,\dots,X_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be ...
7
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0answers
168 views

Recover the field from its Milnor K-groups

For every field $F$, consider $K_n^M(F)$ the $n$-th Milnor K-group of $F$ for each $n \in \Bbb N$, and form the Milnor K-ring $K^M(F)=\oplus_{n \geq 0}K^M_n(F)$. For instance, $K_1(F)=F^{\times}$. ...
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0answers
159 views

Ax theorem for separably closed fields

For the algebraically closed fields a theorem of Ax states that any injective polynomial map from $K^n$ to $K^n$ where $n\in \mathbb{N}$ and $K$ an algebraically closed field, is bijective. Is there ...
3
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0answers
66 views

Algorithm to compute minimal polynomials

Suppose $L/K$ is a finite Galois extension of fields of degree $n$. Suppose that we know an irreducible polynomial $f\in K[x]$ such that $L\cong K[x]/(f)$. Suppose also that we know the Galois group ...
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1answer
113 views

Special elements of the Cremona group

After asking this MO question, I wish to ask about the following special case: Let $f$ be a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$. Is it possible to ...
6
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0answers
69 views

On the isomorphism of the lattices of submodules of certain free modules

Let $K,L$ be two finite extensions of the $p$-adic field $\mathbb{Q}_p$ of the same degree. Let $\mathcal{O}_K$ and $\mathcal{O}_L$ be the ring of integers of these two fields, and let $\mathcal{O}_K^...
5
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0answers
122 views

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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0answers
78 views

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 ...
7
votes
1answer
406 views

What is the topology on the set of field orders

Inspired by this question I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)? For example for the function field $\...
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0answers
108 views

Frobenius twist of a field

Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
6
votes
1answer
151 views

Transcendent basis for the field of multisymmetric functions

It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is, rational function in variables $x_{11}, \ldots, x_{1m}, \ldots, x_{n1}, \ldots, x_{nm}$ ...
0
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2answers
240 views

Splitting field of an intermediate field

Consider the following 'wrong' question. Let $f(x) \in F[x]$ be an irreducible polynomial in a polynomial ring of a field $F$. Let $L$ be the splitting field of $f(x)$ over $F$. Assume that $L$ is a ...
2
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0answers
111 views

Sums of squares in fields

Which fields $k$ have the property that any sum of squares is a square ? Are there elegant characterizations and/or classifications known for this type of field ? And what if we replace "fields" by "...
10
votes
1answer
310 views

The real numbers as a wreath product?

In Faltin-Metropolis-Ross-Rota's [FMRR] paper The Real Numbers as a Wreath Product [Adv. Math. 16(3), 278-304 (1975)], the real numbers are constructed as a quotient of a certain subset of the ring of ...
1
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1answer
104 views

Collineations of projective spaces and isomorphisms of fields

For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for ...
3
votes
0answers
95 views

Does local Langlands say anything about the isomorphism class of the absolute Galois group?

I have heard some people claim that the Local Langlands Correspondence over $\mathbb{Q}_p$ (when it is known) is a deep theorem about representations of the absolute Galois group of $\mathbb{Q}_p$. My ...
0
votes
1answer
150 views

p-adic field extension of degree 2n without a subfield of degree 2?

I need an example of a p-adic field extention $L/F$ of degree $[L:F]=2n$ without a subfield $K\subset L$ of degree $[K:F] = 2$.
3
votes
1answer
136 views

A field having higher Witt vectors as completions

Given a finite field $F$ there is a unique (up to isomorphism) absolutely unramified complete DVR $W(F)$ of mixed characteristic that has $F$ as its residue field. Fix a positive integer $n$. Does ...
17
votes
2answers
520 views

Are radicals dense in the real closure of an ordered field?

Let $F$ be an ordered field and let $R$ denote its real closure. It is well-known that $F$ is cofinal in $R$, but not necessarily dense. For example, consider $F=\mathbb{R}(\omega)$ with the order ...
2
votes
0answers
61 views

Extension of valuations and completion

Let $L/K$ be a finite extension of fields where $K$ is a number field. Let $v$ be a valuation of $K$ and denotes by $K_v$ its completion for this valuation. One denotes $\tilde v$ be the valuation $...
10
votes
1answer
240 views

Derivative of an algebraic power series in positive characteristic

Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$. Indeed let $P(X)=\...
2
votes
0answers
71 views

Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...
0
votes
1answer
119 views

Family of zeros of polynomials

Let $k$ be an infinite field and $P(X_1,\dots,X_n)\in k[t][X_1,\dots, X_{n}]$, suppose that there exists a finite field extension $L$ of $k$ such that $P(x'_1,\dots,x'_n)\in L[t]^{*}=L^{*}$ with $x'...
9
votes
1answer
214 views

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 ...
2
votes
1answer
111 views

Geometric regularity for infinitely generated field extensions

Let $k$ be a field. Suppose that for a finite type $k$-algebra $A$, we define two following properties: $A\otimes_k k'$ is a regular ring for all finitely generated field extensions $k\subset k'$. $...
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0answers
99 views

Nature of polynomials of the form $x^n-a$ over finite fields

I state the following theorem from Serge Lang's Book- Algebra(3rd edition). Theorem: Let $k$ be a field and $n$ an integer $\geq$ 2. Let $a\in k, a\neq 0$. Assume that for all primes $p$ such that $...
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0answers
161 views

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 ...
4
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0answers
226 views

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 "...
42
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0answers
863 views

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 ...
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0answers
146 views

Automorphisms of $\mathbb C_p$ with constraints

In Automorphisms of $\mathbb C_p$, K. Conrad showed that there exist uncountably many $\mathbb Q_p$-automorphisms of $\mathbb C_p$. I have a quite similar question. Let $(a_n)_{n\in\mathbb N}$ and $(...
15
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2answers
983 views

Biggest Field Of Characteristic $p$

The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every ...
7
votes
1answer
227 views

Absolute value on tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$. Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...
14
votes
1answer
432 views

Sum of subfields of $\mathbb{C}$

Do there exist algebraically closed subfields $F_1, F_2, \dots, F_n$ ($n \geq 2$) of the field of complex numbers such that no $F_i$ is contained in $\bigcup_{j \neq i} F_j$ and $F_1 + F_2 + \dots F_n ...
9
votes
0answers
205 views

Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...

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