**3**

votes

**1**answer

99 views

### Could a non-algebraically closed PAC field be a finite extension of an ordered field?

Is there such an example? Or it is known that a pseudo algebraically closed field which is a finite extension of a formally real field is algebraically closed?

**4**

votes

**1**answer

327 views

### Field extension of fields [closed]

Is the field of real numbers $\mathbb{R}$ a finite extension of some subfield $k\subset \mathbb{R}$?

**32**

votes

**1**answer

657 views

### A Topology such that the continuous functions are exactly the polynomials

(I originally asked this question on Math.SE, where it received a lot of attention, but no solution.)
Which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous ...

**12**

votes

**2**answers

546 views

### To what extent can fields be classified?

The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and ...

**2**

votes

**1**answer

165 views

### English translation of Steinitz 1910?

Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042

**1**

vote

**0**answers

34 views

### Degree of factor of resolvent

As always with my questions this is not at research level, but the assertion is made in a research paper, plus no one's been able (or willing) to answer it over at MSE. Here is the original question, ...

**3**

votes

**2**answers

273 views

### Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?

**2**

votes

**0**answers

73 views

### Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$. Is then $F\cap k[x_1,\dots,x_n]=k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$?

Let $F\subseteq k(x_1,\dots,x_n)$ be a subfield with $k\subseteq F$. I know that $F=k(\psi_1,\dots,\psi_r)$ for rational functions $\psi_i\in k(x_1,\dots,x_n)$. I'm interested in the intersection ...

**2**

votes

**0**answers

149 views

### purely non-algebraic extension that is not separable

Can you give an example of a field extension $k\subseteq K$ such that, every element of $K$ is transcendental over k and $K$ is not separable over $k$?

**3**

votes

**1**answer

120 views

### Recognizing etale covers on the level of function fields

Let $X$ be a connected, integral curve over a field $k$, and let $Y \rightarrow X$ be a finite etale cover. Corresponding to this cover there is a finite extension of function fields $k(Y): k(X)$.
...

**3**

votes

**1**answer

365 views

### Fields whose embeddings into the complex numbers are invariant under complex conjugation

Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...

**2**

votes

**0**answers

113 views

### Multiplicative groups in field extensions

If we let $K$ be a number field, thank to the fact that we can extend it integer ring to an UFD where the group of units is finitely generated, we can show that
$K^\ast\cong K^\ast_{tor}\times ...

**7**

votes

**0**answers

222 views

### Simple automorphism groups of field extensions of infinite transcendence degree

Let $k$ be an algebraically closed field and let $K/k$ be a field extension of infinite transcendence degree where $K$ is algebraically closed. Is it true that $\mathrm{Aut}_k(K)$ is a simple group?
...

**1**

vote

**0**answers

122 views

### Finite extensions of residue fields of Henselian DVRs

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...

**0**

votes

**1**answer

141 views

### Sums of Squares and Totally Positive Numbers

In Van der Waerden B L. Algebra Vol.I[M]. Springer, 2003, Pro. Waerden announced in page 256 that if an element $\gamma$ of a formally real field K is not a sum of squares, there exist an ordering of ...

**7**

votes

**4**answers

930 views

### The “interplay” between additive and multiplicative structure in a field

A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
...

**1**

vote

**2**answers

332 views

### Subfield of rational function field and which is not a rational function field

Let $K = k(x_{1}, x_{2},...,x_{n}), n\geq 2, k$ is a field. Is there exist a subfield of $K$ which is not a rational function field? Thanks.

**1**

vote

**0**answers

65 views

### “almost prime” elements in perfect Hahn field

Let $K$ be the field $\mathbb{F_p}^{alg}((\mathbb{Q}))$ (field of Hahn series over
$\mathbb{F_p}^{alg}$ and with value group $\mathbb{Q}$).
Is there elements $x$ of $K$ which are "almost prime" that ...

**2**

votes

**1**answer

190 views

### Multiple eigenvalues over imperfect fields

Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\alpha(A)$ for the ...

**1**

vote

**0**answers

96 views

### Complementation in an extension field

If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is ...

**4**

votes

**1**answer

361 views

### How does an irreducible polynomial of prime power order split over an extension of prime power degree

I asked this question in a similar form on math.se here, where it has been unanswered for a little over a week some work on it can be found there. The motivation for this question was another ...

**0**

votes

**0**answers

167 views

### Pseudo-cauchy sequence and valuation

Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + ...

**5**

votes

**1**answer

464 views

### Algorithm for determining whether two polynomials have the same splitting field

This question asks how to tell whether two cubic polynomials with coefficients in $\mathbb{Q}$ have the same splitting field. There are several answers to the question, but they don't include proofs. ...

**1**

vote

**1**answer

230 views

### Henselization of valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks

**13**

votes

**2**answers

623 views

### The Surreal numbers satisfy all the field axioms except that its elements constitute a proper class. Is it safe to call it a field?

Do all the field theorems apply to surreal numbers? If fields were redefined so that their elements were allowed to come from an arbitrary class, would the theory look different to an algebraist?

**9**

votes

**1**answer

360 views

### Sum of commuting semisimple operators

Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a $T$-invariant complement(for algebraically ...

**0**

votes

**2**answers

250 views

### a problem about field extension [closed]

Let K and L are fields,L is a sub field of K,and L is isomorphic to K,whether can we get K=L?If true,how to prove? Thanks.

**16**

votes

**0**answers

394 views

### Algebraic closure as a fibrant replacement?

Emil Artin's construction of the algebraic closure of a field $K$ is as follows. Let $K_{0} = K$, and inductively let $\{x_f\}$ be a set of indeterminates indexed by the irreducible $f$ in one ...

**0**

votes

**1**answer

262 views

### Algebraic closure and GIT

Does one need to work over an algebraic closed field in ordre to construct GIT quotients à la Mumford?
If yes, why?

**0**

votes

**0**answers

124 views

### Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ is abelian over $\mathbb{Q}_{p}$?

Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the
field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ `
is abelian over $\mathbb{Q}_{p}$?

**4**

votes

**0**answers

358 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

202 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

399 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

228 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 ...

**1**

vote

**0**answers

88 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

422 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 ...

**37**

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

264 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

224 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

195 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

195 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

151 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

237 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

214 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

453 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

209 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

523 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

990 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

**2**answers

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

**3**answers

517 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 ...