All Questions
Tagged with ra.rings-and-algebras fields
61 questions
0
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61
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$\mathcal{R}$ is finite over $L_0[e,A]$
Let $\sigma:L \longrightarrow L$ an automorphism of infinite order, where $L_0 \subset L$ is its fixed field. Let $R$ be a commutative subring of $L\{\sigma\}$. Let
\begin{equation}
\mathcal{R}=\...
- 69
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
0
votes
1
answer
222
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On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
10
votes
1
answer
243
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If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
0
votes
1
answer
219
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Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 2,837
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votes
1
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473
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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 $R$ ...
- 2,837
1
vote
0
answers
107
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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)=\...
- 2,837
5
votes
1
answer
344
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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{...
- 53
2
votes
0
answers
81
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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 ...
- 41.8k
21
votes
1
answer
1k
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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 ...
- 63.9k
7
votes
1
answer
161
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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\...
- 2,811
4
votes
1
answer
198
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 ...
- 4,547
1
vote
0
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233
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 ...
- 157
2
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0
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98
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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 ...
- 1,755
5
votes
0
answers
302
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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 ...
- 32.4k
7
votes
1
answer
284
views
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 ...
- 237
1
vote
1
answer
285
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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) \...
- 3,034
7
votes
1
answer
817
views
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 ...
- 2,017
13
votes
0
answers
571
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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; ...
- 32.4k
43
votes
18
answers
5k
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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 ...
1
vote
2
answers
368
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How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]
Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$.
Let $G$ be the group ...
user178596
2
votes
0
answers
227
views
Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal
Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
- 81
1
vote
0
answers
73
views
Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?
The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.
Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
- 10.1k
7
votes
1
answer
767
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Is the Euler–Mascheroni constant an EL-number?
This question is based on Chow - What is a closed-form number?.
The author of the linked paper had proposed a plausible definition of "elementary numbers" (which he calls "EL-numbers&...
- 10.1k
4
votes
0
answers
197
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Are there any 1-decidable algebraic extensions of $\mathbb{Q}$ which are not decidable?
A model $M$ is decidable if the set of all first-order formulas which are true in $M$ is a recursive set. And a model is $1$-decidable if the set of all existential formulas which are true in $M$ is ...
- 4,587
4
votes
1
answer
287
views
What is the lowest complexity definition of $\mathbb{Z}$ in an infinite algebraic extension of $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
- 4,587
6
votes
1
answer
518
views
What is the Galois group of one ultrapower over another ultrapower?
Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$?
...
- 4,587
1
vote
1
answer
101
views
When is a infinite transcendence-degree rigid fields fixed by a finite extension?
A field is rigid if it has no nontrivial automorphisms. Let $F$ be a rigid field which has infinite transcendence degree over $\mathbb{Q}$, and let $E$ be a finite extension of $F$. Then my question ...
- 4,587
2
votes
0
answers
68
views
division of polynomials [closed]
Consider the set of polynomials $k[x^1,\cdots,x^n]$ over any field $k$. Now, given $p,q\in k[x^1,\cdots,x^n]$, what are the necessary and suficient conditions in order to q divide p? That is: $\exists ...
- 179
0
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0
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140
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Field of algebraic functions
We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...
- 1
3
votes
1
answer
252
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 ...
- 6,962
10
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0
answers
219
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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, $K_1(F)=F^{\times}$.
...
- 6,582
1
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0
answers
146
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Characterisation of projective modules over tensor products of fields
Let $k$ be a field and $L_1$ and $L_2$ finite field extensions of $k$.
Let $A:=L_1 \otimes_k L_2$ as an algebra.
Question:
Given a finitely generated $A$-module $M$, do we have that $M$ is ...
- 26.5k
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
6
votes
1
answer
449
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Octonion algebras over $\mathbb{F}_p(t)$
In their book Octonions, Jordan Algebras and Exceptional groups
Springer and Veldkamp have a subsection called 'Classification over special fields' in which they describe the number of division and ...
- 2,483
15
votes
1
answer
524
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Non standard extension of real numbers via nonprincipal ultra filters
Assume That $U,V$ are two filters on the natural number $\mathbb{N}$.
We say that $U$ is equivalent to $V$ if there is a bijection $\phi: \mathbb{N} \to \mathbb{N}$ such that $\tilde{\phi}(U)=...
- 356
30
votes
0
answers
1k
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Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?
Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
- 1,742
4
votes
1
answer
505
views
Division ring on a field
Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$.
In the field extensions we know that $a^2-2ab+b^2=0$ if and ...
- 221
-1
votes
1
answer
246
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Maximal commutative subrings of the endomorphism ring of a vector space
Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
- 1,959
14
votes
2
answers
1k
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Obstructions for a group to be the multiplicative group of a field [duplicate]
It is well known that every finite multiplicative subgroup of a field is cyclic.
I somehow got interested in a possible reverse implication:
Assume we have an abelian group $G$ whose every finite ...
- 6,741
9
votes
1
answer
522
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Division algebras over extension fields / reducibility of $G$-modules
Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
- 469
1
vote
0
answers
108
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(Affine) Schemes and the point of view of morphisms with values in a field
Let $A$ be a commutative ring with unity. If $\varphi_1 : A \rightarrow K_1$ and $\varphi_2 : A \rightarrow K_2$ are ring morphisms from $A$ to fields, $\varphi_1$ and $\varphi_2$ are said to $1$-...
- 255
1
vote
0
answers
228
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How can I decode efficiently a triple-error-correcting binary BCH code?
In a given $\mathrm{BCH}(N,K)$, $T=3$ code over $\mathrm{GF}(2^m)$, there are ways to find the error locations in a given $N$-bit codeword directly from the syndromes without going through the normal ...
- 87
0
votes
1
answer
164
views
On separable field extensions [closed]
Let $F\subseteq K$ be a finite separable field extension with $a_1,..., a_n$ an $F$-basis for $K$. Is it true that the matrix $A := [\mbox{tr}(a_ia_j)]$ is non-singular ?
- 447
3
votes
0
answers
141
views
Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$. Is then $F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$?
Let $F\subseteq k(x_1,\dots,x_n)$ be a subfield with $k\subseteq F$. I know that $F=k(\psi_1,\dots,\psi_r)$ for rational functions $\psi_i\in k(x_1,\dots,x_n)$. I'm interested in the intersection $F\...
- 339
1
vote
0
answers
107
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Complementation in an extension field
If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is finite....
- 11.4k
4
votes
0
answers
2k
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Existence of algebraic closure and Axiom of choice [duplicate]
Possible Duplicates:
Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
algebraic closure of commuting pairs of matrices
we need zorn's ...
- 1,788
1
vote
1
answer
542
views
Is there a subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is a finite algebraic extension of $F$. [duplicate]
Possible Duplicate:
Examples of algebraic closures of finite index
The question is in the title.
I can prove that if such field $F$ exist then the extension $\mathbb{R}/F$ cannot be of degree $2$...
- 1,701
4
votes
1
answer
256
views
Heisenberg-type groups over rings with involution
Hello everyone!
In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:
Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...
- 3,186
20
votes
2
answers
683
views
non-isomorphic stably isomorphic fields
Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)?
Q2: Do we have a sufficient criterion for ...
- 7,609