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2
votes
1answer
187 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
0answers
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
1answer
336 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
0answers
134 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
1answer
433 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
1answer
219 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
2answers
602 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?
4
votes
1answer
244 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 complement(for algebraically closed fields ...
0
votes
2answers
245 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
0answers
365 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
1answer
254 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
0answers
118 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
0answers
342 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
1answer
191 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
1answer
380 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
0answers
218 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
0answers
87 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
0answers
355 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 ...
34
votes
2answers
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
2answers
255 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
2answers
212 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
1answer
190 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
1answer
188 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
0answers
148 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
2answers
233 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
1answer
197 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
2answers
437 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
207 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
506 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
956 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
504 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
277 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
78 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
250 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
486 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
408 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
534 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
192 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
580 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
936 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
762 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
609 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
428 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
412 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
220 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
582 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

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
464 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 $F^{\ast}$?