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votes

**4**answers

949 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

**1**answer

641 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

**1**answer

421 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

**2**answers

889 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

**1**answer

580 views

### Converse to Banachâ€™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

**1**answer

194 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

**1**answer

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 ...

**6**

votes

**1**answer

510 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

**4**answers

702 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

**1**answer

395 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

**1**answer

156 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

**2**answers

449 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

**2**answers

487 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

**2**answers

471 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

**2**answers

668 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

**7**answers

5k 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

**12**answers

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

**2**answers

905 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

**0**answers

918 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

**2**answers

720 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 $L/K$ ...

**13**

votes

**1**answer

548 views

### Galois theory: Generalization of Abelâ€™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

**1**answer

303 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

**1**answer

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?

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votes

**0**answers

352 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 ...

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votes

**3**answers

1k 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

**1**answer

412 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

**1**answer

472 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 ...

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vote

**2**answers

391 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 ...

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votes

**4**answers

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 ...

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votes

**2**answers

469 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

**3**answers

876 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

**4**answers

916 views

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

Are the computable numbers one example?

**3**

votes

**3**answers

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

**1**answer

608 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 ...

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votes

**2**answers

859 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

**5**answers

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

**0**answers

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 ...

**2**

votes

**1**answer

666 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 ...

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votes

**3**answers

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

**1**answer

859 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

**1**answer

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

**3**answers

395 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 ...

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votes

**2**answers

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 ...

**3**

votes

**0**answers

124 views

### 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 ...

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votes

**3**answers

808 views

### 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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vote

**2**answers

677 views

### 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 the center K and [L : K] < infinity?
I think, but ...

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votes

**0**answers

320 views

### 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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votes

**2**answers

2k views

### 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 ...

**2**

votes

**1**answer

537 views

### 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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**0**answers

9k views

### 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 ...