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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18
votes
2answers
455 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 ...
1
vote
0answers
213 views

Automorphism groups of fields

Hi there, is there a classification/characterization of fields K for which the automorphism group Aut(K) has the property that |Aut(K)| < |K| (e.g. finite fields, the rationals and reals) ? What ...
5
votes
3answers
528 views

Octic family with Galois group of order 1344?

Does the octic, $\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$ for any constant n have Galois group of order 1344? Its discriminant D is a perfect square, $D = ...
6
votes
1answer
993 views

Mysterious property of $\mathbb{Q}$

Hi, I am currently working through the paper by Bousfield and Gugenheim on rational homotopy theory, and have come to a point where they show why it is important to work over $\mathbb{Q}$, and not ...
11
votes
2answers
1k views

Automorphisms of $\mathbb{C}$

Is it true that $G_{\mathbb{Q}}$, the absolute Galois group of $\mathbb{Q}$, is a subgroup of $Aut(\mathbb{C})$ ? Or a simpler question: can any automorphism of $\overline{\mathbb{Q}}$ be extended to ...
2
votes
3answers
519 views

Field constructions

If $F$ is a field of characteristic $p$ prime, how can one create a field $K$ such that $K$ is created from $F$ (either by modding out or by taking a product which includes $F$ or by some other method ...
4
votes
1answer
284 views

Finding purely transcendental parts of field extensions

If we have a field $K$ such that $K\cong K(t)$ (i.e. it is isomorphic to the field you get if you adjoin one transcendental) then is there necessarily a subfield $L\lt K$ such that $L\ncong L(t)$ and ...
1
vote
0answers
80 views

Explicit expression for multivariable meromorphic series

Let $K$ be a field. For $m>1$, elements in the ring of power series in $m$ variables over $K$, i.e., $K[[X_1, \cdots, X_m]]$ look like - $$ \displaystyle\sum_{(i_1, \cdots, i_m)\in (\mathbb ...
3
votes
2answers
256 views

Classes of fields and Cantor-Schröder-Bernstein

In what classes of fields does CSB hold? That is to say, in what classes of fields is it true that if there exist embeddings $F\to K$ and $K\to F$ then $F$ and $K$ must be isomorphic? I know this ...
1
vote
1answer
505 views

Is this function field extension a Galois extension ?

Setting and question Let $X$ be a variety over an algebraically closed field of null characteristic, and let $C$ be a (regular if you want) curve included in $X$. Consider $X'$ the normalization of ...
6
votes
1answer
443 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 ...
3
votes
2answers
610 views

Algebraically closed fields with proper maximal subfields

Is there a classification of the algebraically closed fields that have maximal proper subfields ? And if an algebraically closed field has a maximal proper subfield, is that subfield unique ? ...
2
votes
0answers
194 views

is exponential diophantine over Qp

Thanks to Matiyasevic, we all know that exponential is diophantine over the integers. Also, thanks to transcendental number theory, we know that exponential is not diophantine over the rationals. So ...
6
votes
1answer
689 views

Ring of Witt vectors and p-adics

This is probably an easy question, but I'm not able to figure it out. Are the following the same: Field of fractions of the ring of Witt vectors over the algebraic closure of $\mathbb{F}_p$ ...
10
votes
3answers
961 views

Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?

Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$. Can we always find such an irreducible ...
1
vote
3answers
1k views

Constructive proof of algebraic elements forming a subfield

Let $F \leqslant E$ be a field extension. If $a, b \in E$ are algebraic over $F$ then $a+b$ and $ab$ are algebraic as well. There is an short proof of this using the extension $E(a,b)$: $[E(a,b):E]$ ...
10
votes
2answers
654 views

Constructing the surreal numbers as iterated Hahn series

A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with ...
3
votes
1answer
466 views

Is the transcendence degree of a domain over a subfield the same as that of the fraction field of that domain?

Consider the inclusion $k\subset A$ of the field $k$ in the domain $A$ and the fraction field $K=Frac(A)$ of $A$. Obviously if a family $(a_i)_{i\in I}$ of elements $a_i \in A$ is algebraically ...
10
votes
3answers
1k views

Which polynomials are determinants of a symmetric matrix with linear entries?

Let $k$ be a field. Can each degree $n$ polynomial $P(t) \in k[t]$ be written as the determinant of the matrix $A + tB$, where $A$ and $B$ are two symmetric $(n \times n)$-matrices with entries in ...
5
votes
2answers
424 views

Given 2 towers of fields, when are these fields isomorphic?

Let $F_0 \subset F_1 \subset F_2 \subset \cdots$ and $K_0 \subset K_1 \subset K_2 \subset \cdots$ be two towers of fields. Also, let $F = \cup_{i=0}^\infty F_i$ and $K = \cup_{i=0}^\infty K_i$. Now ...
4
votes
1answer
229 views

Does the weak approximation theorem hold for general topological fields?

The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
4
votes
3answers
613 views

Field with cyclic product group

If a field has a cyclic multiplicative group, is it necessarily finite?
2
votes
4answers
1k views

When a group ring is a local ring [closed]

Hi there, I'm stuck with my undergraduate thesis on the following proposition: If $k$ is a field of characteristic $p > 0$ and $G$ is a finite $p$-group, then the group ring $kG$ is local. In ...
11
votes
1answer
475 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 $F^{\ast}$?
7
votes
4answers
1k views

Finite dimensional vector spaces over a complete but not-necessarily-valued field

I'm essentially reopening this old question of Ricky Demer which was never fully answered. Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and ...
13
votes
1answer
695 views

Is every algebraic extension of a field of absolute transcendence degree one a separable extension of a purely inseparable extension?

Any decent course on field theory will state that in characteristic $p$ an extension of fields $k\subset K$ canonically decomposes as the tower $k\subset K_{sep}\subset K$ with $K$ purely inseparable ...
1
vote
1answer
425 views

Does “all points rational” imply “constant” for this “cubic” curve over an arbitrary field?

Let $\mathbb{K}$ be an arbitrary field with a subfield $\mathbb{F}$ of index 2. Let $a,b\in\mathbb{K}[X]$ be univariate non-vanishing polynomials over $\mathbb{K}$ of degree $\leq 3$ each. Edit: Due ...
6
votes
2answers
964 views

Why isn't the perfect closure separable?

This question has escalated from math.stackexchange. I'm doing this because it has been a while since the question has been open, receiving no satisfactory answers, even when subject to a bounty. I ...
11
votes
1answer
629 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| := ...
2
votes
1answer
201 views

Isomorphism problem for finite dimensional central division algebras over a function field in two indeterminates.

Let K be the fraction field of C[x,y] where C denotes the complex numbers. Suppose D and E are two central division algebras over K of degree n, i.e. dim(D)=dim(E)=n^2. Is there any natural criterium ...
2
votes
1answer
326 views

On the Separability of Certain Extensions of Fields.

Hi, I made this question a couple of weeks ago. The question arose while looking for a criterion of separability for extensions of fraction fields $K(A)\to K(B)$ induced by a faithfully flat ...
9
votes
1answer
567 views

an algebraically closed field definable in a real closed field

Is it true that an algebraically closed field $k$ definable (in the model theory sense) in a real closed field $\mathcal R$ is the algebraic closure $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$? ...
4
votes
4answers
753 views

What are the lengths that can be constructed with straightedge but without compass?

Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. ...
5
votes
1answer
481 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 ...
1
vote
1answer
159 views

Behaviour of Primes under Regular Coefficient Extensions

Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
8
votes
2answers
452 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
2answers
509 views

Valuations and separable extensions

Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable? I have convinced myself that (for a positive answer) ...
3
votes
2answers
474 views

n-th roots of Pythagorean numbers

Let $F$ be the field ${\mathbb Q}(i)\subset \mathbb C$ and let $T\subset F$ be the set of all elements of complex absolute value 1. Let $n$ be a natural number $\ge 2$ and let $\mu_n(T)\subset\mathbb ...
2
votes
2answers
735 views

Countable Fields with No Countable Extension

Let $\mathscr{S}$ be the set of all countable subfields of $\mathbb{C}$. Clearly, $\mathscr{S}$ is a partially ordered set under inclusion, and if $K_1\subseteq K_2 \subseteq \cdots$ is an ascending ...
26
votes
7answers
8k views

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 ...
15
votes
12answers
2k views

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 ...
26
votes
2answers
959 views

If a field extension gives affine space, was it already affine space?

Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the algebraic closure $\overline F$. Does it ...
20
votes
0answers
975 views

Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
11
votes
2answers
867 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 $L/K$ ...
14
votes
1answer
558 views

Galois theory: Generalization of Abel’s Theorem? (Better version!)

(Unintentionally I have previously asked a similar and perhaps in itself not uninteresting question Galois theory: Generalization of Abel's Theorem? but this is what I originally had in mind.) ...
9
votes
1answer
314 views

Set of maximal subfields not containing particular elements.

Instead of extending a field, by adjoining a new element, consider what happens if we remove an element or elements. This started as a question on math.SE Field reductions where Pete L. Clark ...
8
votes
1answer
1k views

Is every field the field of fractions of an integral domain?

Is every field the field of fractions of an integral domain which is not itself a field? What about the field of real numbers?
2
votes
0answers
357 views

What do you call an algebraic element with the property that the generated field extension is normal?

Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you ...
22
votes
3answers
2k views

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 ...
3
votes
1answer
420 views

Function Fields of Real Varieties

Let $V$ be a geometrically irreducible and reduced scheme defined over the real numbers, and let $K = K(V)$ be its function field. If $V$ does not have any real points, is it true that $K$ is not ...