**4**

votes

**0**answers

374 views

### How arithmetical is algebraic exponentiation?

Suppose $K$ is an exponential real closed field, i.e. there is an isomorphism, say exp, between the additive group of $K$ and the multiplicative group of its positive elements.
Assume further that $Z$ ...

**3**

votes

**1**answer

213 views

### Nondegenerate involutions on the complex numbers — always just conjugation?

Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum_i a_i a_i' = 0 \Leftrightarrow \forall i \ a_i=0$, where this sum is finite.
Must this ...

**0**

votes

**1**answer

420 views

### Does this isomorphism between Galois groups hold for transcendental extensions?

Suppose $L/K$ and $M/K$ are algebraic extensions of a field $K$, such that $L\cap M=K$, and $L/K$ is a normal extension. It is well-known that, with these conditions, we have:
$$\text{Gal}(L/K)\cong ...

**1**

vote

**0**answers

253 views

### The field $\mathbb{Q}(\cos \frac {2\pi} {n})$ [closed]

Let $x$ be $\cos \displaystyle \frac {2\pi} {n}$ for some natural number $n$.
Then is there an integer $n$ such that $\mathbb{Q}(x^2+x)\neq \mathbb{Q}(x)$?
I also would like to know if there is some ...

**3**

votes

**1**answer

126 views

### when is the property “being algebraically maximal” a first order property ?

A valued field is said to be algebraic maximal if all its algebraic extension have either a bigger value group or a bigger residue field.
Do you know for which field this is a first order property ?
...

**1**

vote

**0**answers

501 views

### Existence of algebraic closure and Axiom of choice [duplicate]

Possible Duplicates:
Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
algebraic closure of commuting pairs of matrices
we need ...

**38**

votes

**2**answers

1k views

### Isomorphic general linear groups implies isomorphic fields?

Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it ...

**7**

votes

**2**answers

276 views

### Solving the Field Membership Problem using Grobner Bases

Is there an easy way to determine whether a set of elements in a field generates the whole field or only a subfield?
Specifically, I have a subfield of $k(x,y)$ described in terms of a canonical set ...

**4**

votes

**2**answers

236 views

### an example of a strictly superstable field

It is well-known that the theory of separably closed fields of some fixed positive characteristic and degree of imperfection is stable but not superstable. By a result of Cherlin and Shelah a ...

**0**

votes

**1**answer

200 views

### Is there a subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is a finite algebraic extension of $F$. [duplicate]

Possible Duplicate:
Examples of algebraic closures of finite index
The question is in the title.
I can prove that if such field $F$ exist then the extension $\mathbb{R}/F$ cannot be of ...

**4**

votes

**1**answer

201 views

### Heisenberg-type groups over rings with involution

Hello everyone!
In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:
Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...

**1**

vote

**0**answers

160 views

### Extending commuting endomorphims of a complete discrete valued field to the algebraic closure?

Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?

**9**

votes

**2**answers

239 views

### Classifying space of the unit circle of a valued field

Let $k$ be a field with absolute value $|\cdot |$ and let $S^1(k) := \lbrace x \in k \mid |x|=1\rbrace \le k^\times$. Since the absolute value definies a metric on $k$, $S^1(k)$ is a topological ...

**6**

votes

**1**answer

220 views

### When is a valued field second-countable?

Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial).
The valuation $v:K^{\times}\to\Gamma$ ...

**18**

votes

**2**answers

493 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

**0**answers

217 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

**3**answers

539 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

**1**answer

1k 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 ...

**12**

votes

**2**answers

2k 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

**3**answers

525 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

**1**answer

294 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

**0**answers

83 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

**2**answers

262 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

**1**answer

533 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

**1**answer

465 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

**2**answers

677 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

**0**answers

195 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

**1**answer

789 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

**3**answers

979 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

**3**answers

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

**2**answers

686 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

**1**answer

494 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

**3**answers

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

**2**answers

433 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

**1**answer

232 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

**3**answers

639 views

### Field with cyclic product group

If a field has a cyclic multiplicative group, is it necessarily finite?

**2**

votes

**4**answers

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

**1**answer

489 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

**4**answers

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

**1**answer

740 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

430 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

**2**answers

1k 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 ...

**12**

votes

**1**answer

649 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

**1**answer

204 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

330 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

**1**answer

577 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

762 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

**1**answer

490 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 ...

**2**

votes

**1**answer

168 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

457 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 ...