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].
562
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Methods of showing an element is / is not in a field
Let $K$ be a field, $\alpha\in\bar{K}$, and $L/K$ a finite extension. How can we determine whether $\alpha\in L$, preferably in as much generality as possible?
Of course, there may be special cases ...
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Primitive element theorem without building field extensions
Is there are nice way to prove the primitive element theorem without using field extensions?
The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over $F$...
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Analogue of a ring extension splitting in the Kummer case
Background (the Kummer extension case)
Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested in $R=...
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Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?
My question is a doubt I had in the last point to the first answer to this MO question - "Algebraic" topologies like the Zariski topology?
Can one associate a Riemann surface to any ...
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Finite extensions of field of rational functions in one variable
Let $K=F(x)$, where $x$ is transcendental over $F$ and $F$ is an algebraically closed field. Does there exist a non-commutative division algebra $L$ with center $K$ and $[L:K]<\infty$?
I think, ...
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Point of confusion in "Topological Representations of Algebras"
Background
I'm reading the article "Topological Representations of Algebras" by Arens, Kaplansky. In the proof of Theorem 6.1 we have the following situation: $X$ is a Stone space, $X_\alpha$ is a ...
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uncountable algebraically closed field other than C ?
Is there any "well-known" algebraically closed field that is uncountable other than $\mathbb{C}$ ?
The algebraic closure of $\mathbb{C}(X)$ would work, but is it meaningful, is this field used in some ...
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Surjectivity of bilinear forms.
It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ ...
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Ultrafilters and automorphisms of the complex field
It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that $\vert\...
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Sum of n-th roots is rarely rational
Let $m,n$ be positive integers, and $\displaystyle \Phi_{m,n}~:~ {\mathbb{R}_+^*}^m \to \mathbb{R}_+^*, \ \ \ (x_1,x_2, \ldots , x_m) \mapsto \sum_{k=1}^m \sqrt[n]{x_k}$.
Clearly for $m=1$ if for all ...
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Is -1 a sum of 2 squares in a certain field K?
Consider the field of fractions $K$
of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$,
where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables.
Clearly $-1$ is a ...
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Making the fixed field algebraically closed
K is a field and f is an automorphism of K. We're also given that the order of f, as an element of the group Aut(K), is not finite. Is it always possible to find a field L which contains K and an ...
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Fields with trivial automorphism group
Is there a nice characterization of fields whose automorphism group is trivial? Here are the facts I know.
Every prime field has trivial automorphism group.
Suppose L is a separable finite extension ...
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Is a field uniquely determined by its multiplicative group/how much knows K_1 about fields?
As the title says I would like to know if $K_1(k)=k^*$ uniquely determines a field $k$.
For finite fields this is clearly the case, but I suspect it is not ture in general. However I guess cooking up ...
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Is there a natural way to view the proof of Hilbert 90?
I only know of one proof of Hilbert 90, which is very smart if not magical. See for example http://hilbertthm90.wordpress.com/2008/12/11/hilberts-theorem-90the-math/
Does anyone know of a more ...
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answer
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When are intersections of finitely generated field extensions finitely generated?
Let $k$ be a field, and let $E$ and $F$ be fields extending $k$, both contained in some single extension of $k$. If $E$ and $F$ are finitely generated (as fields) over $k$, must $E\cap F$ also be ...
- 6,159
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roots of polynomials with multiple-of-unity coefficients implies algebraically closed?
Suppose a field has roots of all polynomials whose coefficients are 0, 0+1, 0+1+1, 0+1+1+1, 0+1+1+1+1, etc or additive inverses thereof. Is such a field necessarily algebraically closed?
The ...
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For which fields K is every subring of K…?
This question was inspired by
How to prove that the subrings of the rational numbers are noetherian?
which some people found too routine to be of interest. So I have decided to liven things up a bit ...
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Existence of maximal totally ramified extensions of an arbitrary CDVF
Let $K$ be a complete, discretely valued field with (let's say) perfect residue field $k$. We have a unique maximal unramified extension $K^{unr}$ of $K$ and a unique maximal tamely ramified ...
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Slightly weakened / altered concepts of a field
I've heard of at least three slight modifications of the standard concept of field:
meadow, which (according to this paper) is a commutative ring with unit equipped with a total unary operation $x^{−1}...
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What are the possible sets of degrees of irreducible polynomials over a field?
Hopefully this is not too easy an exercise.
Let $F$ be a field. Let $I \subset \mathbb{N}$ be the set of all positive integers $d$ such that there exists an irreducible polynomial of degree $d$ over ...
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Does $\operatorname{Con}\sf(ZF)$ imply $\operatorname{Con}\sf(ZF + \operatorname{Aut}{\bf C = Z/\mathrm 2Z})$?
How many field automorphisms does $\mathbf{C}$ have? If you assume the axiom of choice, there are tons of them -- $2^{2^{\aleph_0}}$, I believe. And what if you don't -- how essential is the axiom ...
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Extensions of fields with lots of symmetry
[Cute question heard elsewhere]
Is there a nice characterization of extensions of fields $K/k$ such that whenever $E/k$ and $E'/k$ are subextensions and $\sigma:E\to E'$ is an isomorphism over $k$, ...
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Do separable and normal have topological meanings for fields?
The terminology would suggest that a separable field extension is so because the resulting field extension has some sort of separable topology, and that a normal extension corresponds to one with a ...
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3
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If K/k is a finite normal extension of fields, is there always an intermediate field F such that F/k is purely inseparable and K/F is separable?
I was feeling a bit rusty on my field theory, and I was reviewing out of McCarthy's excellent book, Algebraic Extensions of Fields. Out of Chapter 1, I was able to work out everything "left to the ...
- 6,722
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An unfamiliar (to me) form of Hensel's Lemma
In his very nice article
Peter Roquette,
History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355,
Fields Inst. Commun., ...
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What is the transcendence degree of Q_p and C over Q?
Is the tr.deg of Q_p over Q 1? and what about C over Q?
- 1,503
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Can the algebraic closure of a complete field be complete and of infinite degree?
Yes, this is yet another "foundational" question in valuation theory.
Here's the background: it is a well known classical fact that the dimension (in the purely algebraic sense) of a real ...
- 64.8k
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1
answer
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On using field extensions to prove the impossiblity of a straightedge and compass construction
Let $z \in \mathbb{C}$. Consider the following statements:
The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$.
There is a field extension $K / \mathbb{...
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Galois group of a product of polynomials
How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
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Irreducible Polynomials in UFD and corresponding Quotient Field
Hello,
"Let $D$ be a UFD and let $F$ be its quotient field. Further let
$f$ be a primitive polynomial of positive degree in $D\left[x\right]$.
From this it follows that that $f$ is irreducible in $D\...
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answer
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Orders of field automorphisms of algebraic complex numbers
Let $Aut(\bar{Q})$ be the automorphism group on the field of algebraic complex numbers. The order of an element $f \in Aut(\bar{Q})$ is the least natural number $n$ (if there exists one) such that $(f)...
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Algebraicity of the completion of a field? Finiteness?
At the end of my 8410 class today (see http://alpha.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question:
Let $(K,|\ |)$ be a normed field,...
- 64.8k
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Which commutative groups are the group of units of some field?
Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...
- 115k
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votes
2
answers
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What is it called if a vector space doesn't have an additive inverse?
so, you have, for any two members of the algebraic structure A and B and any nonnegative real values a, b:
two operations: * and +, such that
a*A + b*A = (a+b)*A is in the structure
A + B = B + A ...
- 598
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3
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Are there as many real-closed fields of a given cardinality as I think there are?
Let $\kappa$ be an infinite cardinal. Then there exists at least one real-closed field of cardinality $\kappa$ (e.g. Lowenheim-Skolem; or, start with a function field over $\mathbb{Q}$ in $\kappa$ ...
- 64.8k
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Split powers of the multiplicative group of a field
Let $K$ be a field, $K^\times$ its multiplicative group and $I$ an infinite set. Is then $(K^\times)^{(I)} \subseteq (K^\times)^I$ a direct summand? If not, is it possible to characterize the fields ...
- 61.5k
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Examples of completions and algebraic closures
It is widely known that the algebraic closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$.
I have read ...
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What is the prime spectrum of a Cauchy series ring?
Let $k$ be a field, and let $| \ |$ be a norm on $k$. The norm induces a metric. To construct the completion $\hat{k}$ as a normed field, the standard recipe is to take the quotient of the ring $\...
- 64.8k
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3
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transcendental Galois theory
Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...
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"Kummerian" fields?
This is sort of a random, spur of the moment question, but here goes:
We define [with apologies to Conan the Barbarian] a field K to be $\textbf{Kummerian}$ if there exists
an index set I, and ...
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When f(x)-a and f(x)-b yield the same field extension
An interesting mathoverflow question was one due to Philipp Lampe that asked whether a non-surjective polynomial function on an infinite field can miss only finitely many values. In my interpretation ...
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Field extension containing the eigenvectors of a Hermitian matrix
Let H be a (finite-dimensional) Hermitian matrix with algebraic numbers for its entries, all of which lie in some minimal field extension of the rational numbers; call this field ℚ(H) for short. ...
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Is the theory of incidence geometry complete?
Consider the basic axioms of planar incidence geometry, which allow us to speak of in-betweeness, collinearity and concurrency. These axioms per se are not complete, since for example, Desargues ...
user2529
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Constructing a convex valuation ring/ordered group of rank $n$
I know at least one method of constructing a convex valuation ring of rank $n$ (but it is rather complicated). What are the easiest methods of doing this? Given a natural number $n$ I want to have a ...
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Examples of algebraic closures of finite index
So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
...
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How much choice is needed to show that formally real fields can be ordered?
Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...
- 64.8k
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votes
4
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What does "linearly disjoint" mean for abstract field extensions?
All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
- 11.1k
5
votes
2
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Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
- 64.8k
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votes
3
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Expressing field inclusions by polynomial equalities on coefficients
Let $A$ be the set of all quadruples $(a_0,a_1,a_2,a_3) \in {\mathbb Q}^4$ such that
the polynomial $P=X^4+a_3X^3+a_2X^2+a_1X+a_0$ is irreducible and if $z$ is any root
of $P$, then ${\mathbb Q}(z)$ ...
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