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].
81
questions
32
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3
answers
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Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, asserting that ...
95
votes
11
answers
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views
Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?
Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
27
votes
7
answers
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In what sense are fields an algebraic theory?
Since there is no "free field generated by a set", it would seem that
1) there is no monad on Set whose algebras are exactly the fields
and
2) there is no Lawvere theory whose models in Set are ...
12
votes
1
answer
755
views
Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?
There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$.
We usually call it $\mathbb{C}$, but by this we impose a ...
168
votes
1
answer
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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\...
132
votes
3
answers
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When is the tensor product of two fields a field?
Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often ...
56
votes
14
answers
20k
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Fantastic properties of Z/2Z
Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
35
votes
3
answers
2k
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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 ...
26
votes
3
answers
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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., ...
20
votes
2
answers
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Is there an irreducible but solvable septic trinomial $x^7+ax^n+b = 0$?
The following irreducible trinomials are solvable:
$$x^5-5x^2-3 = 0$$
$$x^6+3x+3 = 0$$
$$x^8-5x-5=0$$
Their Galois groups are isomorphic to ${\rm D}_5$, ${\rm S}_3 \wr {\rm C}_2$ and
$({\rm S}_4 \...
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, ...
83
votes
2
answers
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Why is differential Galois theory not widely used?
E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
70
votes
10
answers
21k
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Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue ...
43
votes
5
answers
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views
Explicit elements of $K((x))((y)) \setminus K((x,y))$
In an answer to the popular question on common false beliefs in mathematics
Examples of common false beliefs in mathematics
I mentioned that many people conflate the two different kinds of formal ...
29
votes
2
answers
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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?
...
28
votes
0
answers
839
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The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...
28
votes
3
answers
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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 ...
22
votes
5
answers
2k
views
Local inverse Galois problem
It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedean field $K$ has solvable (in fact supersolvable [edit: no!]) Galois group $G$. One sees this by using the ramification ...
16
votes
2
answers
1k
views
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 ...
13
votes
2
answers
2k
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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.
12
votes
0
answers
332
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Metric completion of an algebraically closed field is algebraically closed?
Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed?
We can ...
12
votes
5
answers
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Does k(X) have a k-basis for every set X, without AC?
This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?.
For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
9
votes
3
answers
3k
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Some arithmetic terminology: "universal domain", "specialization", "Chow point"
As a non-connoisseur of arithmetic and arithmetic geometry, I would like to ask about some terminology, which meaning I haven't been able to find out on some books, nor on wikipedia, nor by google.
...
7
votes
1
answer
794
views
Parity of class number of pure cubic fields
A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.
What can be said about the parity (odd or even) of the class number of a pure ...
6
votes
1
answer
727
views
What is the size of the smallest rigid extension field of the complex numbers?
Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case $...
6
votes
1
answer
909
views
Finding all automorphisms of $\mathbb{C}(x,y)$
The group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[x,y]$ is well-known, see, for example, the proof of
Dicks or the proof of
Mckay and Wang.
What can be said about the group of $\mathbb{...
5
votes
1
answer
268
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 ...
4
votes
1
answer
392
views
Parametric Solvable Septics?
Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 + 21x^5 + 35x^3 + 7x + a(7x^6 + 35x^4 + 21x^2 + 1)=0\tag{2}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - ...
97
votes
19
answers
36k
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Collecting proofs that finite multiplicative subgroups of fields are cyclic
I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
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 ...
45
votes
4
answers
7k
views
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 ...
39
votes
2
answers
2k
views
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 ...
29
votes
0
answers
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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(...
24
votes
12
answers
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Constructions unique up to non-unique isomorphism
1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...
22
votes
4
answers
3k
views
Which fields have multiplicative group isomorphic to additive group times Z/2Z?
Let $K$ be a field, $K_{*}$ its multiplicative group and $K_{+}$ its additive group. As Richard Stanley notes in this answer, the only field for which $K_{+} \simeq K_{*} \times \mathbb{Z}/2\mathbb{Z}$...
20
votes
1
answer
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On the solvable octic $x^8-x^7+29x^2+29=0$
The irreducible but solvable octic,
$$x^8-x^7+29x^2+29=0\tag{1}\label{1}$$
was first mentioned by Igor Schein in this 1999 sci.math post (Wayback Machine). This does not factor over a quadratic or ...
20
votes
3
answers
2k
views
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 $\kappa$ ...
17
votes
2
answers
2k
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Are there only two smooth manifolds with field structure: real numbers and complex numbers?
Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - $\mathbb{R}$ ...
16
votes
1
answer
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What "should" be the absolute galois group of a field with one element
As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$.
My question is the following:
How we should think or what should be the "absolute Galois ...
15
votes
1
answer
933
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Theory of C* algebras over other fields than the complex numbers
How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...
14
votes
3
answers
2k
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Intuition for Model Theoretic Proof of the Nullstellensatz
I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
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 ...
12
votes
2
answers
1k
views
Factoring a field extension into one which adds no roots of unity, followed by one which adds only roots of unity
I am asking my question here, since it's been voted up a fair bit on math.SE, but without answers, so it may be harder than I assumed it was.
Can we always break an arbitrary field extension $L/K$ ...
12
votes
1
answer
888
views
Converse to Banach’s fixed point theorem for ordered fields?
Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \...
11
votes
1
answer
675
views
torsion group of the multiplicative group of a field
Let $F$ be any field of zero characteristic, $F^{\ast}$ its multiplicative group and $T$ is the torsion group.
Is it true that $T$ is a direct summand for $F^{\ast}$?
11
votes
2
answers
576
views
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,...
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 ...
10
votes
1
answer
1k
views
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 ...
10
votes
3
answers
814
views
Is every field extension of an ultrafield an ultrafield?
Let $K=\lim(K_{i})$ be an ultrafield (over a non-principal ultrafilter), and let $K\hookrightarrow K'$ be a field extension of $K$.
When the field $K'$ is finite over $K$ it is also an ultrafield by ...
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 ...