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3
votes
1answer
442 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
414 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
222 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
590 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
468 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
978 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
655 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
423 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 ...
5
votes
2answers
907 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
592 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
199 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
320 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
559 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
708 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. ...
3
votes
1answer
405 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
450 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
497 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
472 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
693 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 ...
17
votes
7answers
6k 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
924 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 ...
19
votes
0answers
938 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 ...
9
votes
2answers
730 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
554 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
304 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 ...
6
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
353 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 ...
19
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
415 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 ...
8
votes
1answer
478 views

When are the intermediate fields totally ordered?

The groups whose subgroups are totally ordered by inclusion are easy to classify; they are subgroups of $\mathbb{Z}/p^{\infty} = \text{colim } \mathbb{Z}/p^k$ for some prime $p$, and thus ...
1
vote
2answers
402 views

Weakly initial sets - examples and nonexamples

A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I. The question is then, does Fields have a weakly ...
15
votes
4answers
2k views

FFTs over finite fields?

I'm trying to understand how to compute a fast Fourier transform over a finite field. This question arose in the analysis of some BCH codes. Consider the finite field $F$ with $2^n$ elements. It is ...
7
votes
2answers
471 views

An alternative description of K^*/Nm(L^*)

Is there a nice explicit description for the group $K^*/Nm_{L/K}(L^*)$ for a finite field extension $L/K$? What if for example, $L$ is obtained from $K$ by ajoining an n-th root of some $\alpha \in ...
5
votes
3answers
882 views

Is there a field which is the union of finitely many proper subfields?

Is there a field which is the union of finitely many proper subfields?
2
votes
4answers
964 views

What are examples of ordered fields that do not have the Archimedean Property?

Are the computable numbers one example?
3
votes
3answers
1k views

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. ...
3
votes
1answer
622 views

How do you prove that a field is isomorphic to C(x)?

F is a field and F(x) is the field of rational functions in one variable x over F. Are there some clever ways to prove that some field extension K/F is (or is not) isomorphic to F(x)? One way is to ...
12
votes
2answers
866 views

Is completeness of a field an algebraic property?

Pretty straitforward: If a field has a metric in which it is complete can it have a metric in which it is not complete? By metric I mean field norm of course
22
votes
5answers
1k 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 ...
1
vote
0answers
342 views

Quadratic Solutions

There are quadratic solutions to $x^4+y^4 = z^4$ in $\mathbb{Q} (\sqrt{-7})$. But for equations such as $x^4+y^4 = nz^4$ where $n \in \mathbb{N}, \ n \neq 1$ do there still exist extension fields of ...
3
votes
1answer
685 views

How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...
11
votes
3answers
1k views

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 ...
18
votes
1answer
866 views

What is the ring of integers of the Pythagorean field?

Following Hilbert, we call the complex numbers constructible via compass and straight-edge the field of Euclidean numbers, and the totally real such numbers the field of Pythagorean numbers. (Among ...
2
votes
1answer
190 views

behavior of places of a function field under automorphism

if $P_{1}$ and $P_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P_{1})=P_{2}$. then, does $\deg (P_{1}\cap K(x))=\deg ...
2
votes
3answers
399 views

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 ...
5
votes
2answers
2k views

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 ...