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].
563
questions
4
votes
0
answers
160
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 ...
- 363
2
votes
0
answers
237
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 answer is ...
- 26.1k
7
votes
1
answer
255
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 $\...
- 311
3
votes
0
answers
112
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, $\deg m_\alpha$ ...
- 131
3
votes
1
answer
229
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 ...
- 6,484
3
votes
1
answer
135
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 ...
- 19.4k
10
votes
0
answers
211
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, $K_1(F)=F^{\times}$.
...
- 6,158
1
vote
0
answers
222
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 ...
- 11
3
votes
0
answers
157
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 ...
- 509
0
votes
1
answer
131
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 ...
- 2,783
7
votes
0
answers
147
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^...
- 49.5k
5
votes
0
answers
138
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?
user145520
0
votes
0
answers
125
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 ...
- 2,783
7
votes
1
answer
539
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 $\...
- 10.3k
1
vote
0
answers
217
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
1
answer
227
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 $x_{11}, \ldots, x_{1m}, \ldots, x_{n1}, \ldots, x_{nm}$ ...
- 133
0
votes
2
answers
637
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 ...
- 103
2
votes
0
answers
214
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 "...
- 4,353
12
votes
1
answer
557
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 ...
- 805
1
vote
1
answer
158
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 ...
- 40.9k
9
votes
0
answers
402
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 ...
- 91
2
votes
1
answer
274
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 $[K:F] = 2$.
- 397
3
votes
1
answer
146
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 ...
- 31
18
votes
3
answers
637
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 ...
- 2,846
2
votes
0
answers
78
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 $...
- 3,739
11
votes
1
answer
367
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)=\...
- 3,739
3
votes
0
answers
76
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 $\Omega$ ...
- 3,739
0
votes
1
answer
121
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'...
- 3,432
10
votes
2
answers
475
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 ...
- 101
2
votes
1
answer
143
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 $k\subset k'$.
$...
user140042
1
vote
0
answers
113
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 $...
- 428
1
vote
0
answers
202
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 ...
- 4,353
5
votes
0
answers
322
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 "...
- 2,623
48
votes
0
answers
2k
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 ...
- 38.2k
1
vote
0
answers
163
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 $(...
- 3,739
15
votes
2
answers
1k
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 ...
- 933
7
votes
1
answer
324
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{1}...
- 87
5
votes
0
answers
829
views
How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,063
14
votes
1
answer
500
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 ...
- 151
9
votes
0
answers
293
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 ...
- 191
4
votes
0
answers
177
views
Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?
Prove or disprove:
Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$?
Note that then $G_K \...
- 15.1k
3
votes
0
answers
694
views
Is the intersection of two function fields over finite fields again a function field?
I am interested in the question above. I know that the answer is NO if the base field is for instance $\mathbb{Q}$ (the intersection of $\mathbb{Q}(x^2)$ and $\mathbb{Q}((x-1)^2)$ is $\mathbb{Q}$ ...
- 327
1
vote
1
answer
2k
views
Write the algebra closure of $F_p$ as union of finite fields [closed]
This question follows Field theory by Steven Roman, Chapter 9, Exercise 20.
Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite ...
- 205
3
votes
0
answers
85
views
What is known about the Hopf map for quadratic field extensions?
This question is related to my previous post:
Is this generalization of the Hopf map for quadratic field extensions surjective?
I still would like to know more and, while that post got several votes,...
- 5,011
3
votes
0
answers
149
views
Modal Principles of Field Extensions
In 2007 (with more work done later), J. Hamkins and B. Löwe found that the ZFC provably valid principles of forcing are the assertions of S4.2. In the introduction, they mention field extension as a ...
- 131
1
vote
0
answers
135
views
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 ...
- 26.1k
2
votes
1
answer
148
views
Transitivity of an invariant of finitely generated field extensions
For a finitely generated extension of fields $K/k$, let us define "$S_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with ...
- 1,987
2
votes
1
answer
459
views
Having a separable extension of degree $n$ implies having a Galois extension of degree $n$?
I would like an explanation for the fact stated in the title. To repeat:
Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of ...
5
votes
1
answer
613
views
Eudoxus real numbers
I recently remembered the eudoxus construction of the real numbers.
Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction?
Clearification: The usual ...
- 3,161
1
vote
1
answer
314
views
Formal Laurent and Taylor series
Let us define the field of formal Laurent series over a field $k$ as $K=k((x_1))((x_2))...((x_n))$. The subring of formal Taylor series $R=k[[x_1,...x_n]]$ is embedded in this field. Let us call its ...
- 2,235