Expand all Collapse all | Results 1 - 21 of 21 |
1. CMB Online first
Vanishing of Massey products and Brauer groups Let $p$ be a prime number and $F$ a field containing a root of
unity of order $p$.
We relate recent results on vanishing of triple Massey products
in the mod-$p$ Galois cohomology of $F$,
due to Hopkins, Wickelgren, MinÃ¡Ä, and TÃ¢n, to classical
results in the theory of central simple algebras.
For global fields, we prove a stronger form of the vanishing
property.
Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields Categories:16K50, 11R34, 12G05, 12E30 |
2. CMB Online first
Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds We give an explicit formula for the resultant of Chebyshev polynomials of the
first, second, third, and fourth kinds.
We also compute the resultant of modified cyclotomic polynomials.
Keywords:resultant, Chebyshev polynomial, cyclotomic polynomial Categories:11R09, 11R18, 12E10, 33C45 |
3. CMB Online first
Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials Let $v$ be a henselian valuation of any rank of a field
$K$ and $\overline{v}$ be the unique extension of $v$ to a
fixed algebraic closure $\overline{K}$ of $K$. In 2005, it was studied properties
of those pairs $(\theta,\alpha)$ of elements of $\overline{K}$
with $[K(\theta): K]\gt [K(\alpha): K]$ where $\alpha$ is an element
of smallest degree over $K$ such that
$$
\overline{v}(\theta-\alpha)=\sup\{\overline{v}(\theta-\beta)
|\ \beta\in \overline{K}, \ [K(\beta): K]\lt [K(\theta): K]\}.
$$
Such pairs are referred to as distinguished pairs.
We use the concept of liftings of irreducible polynomials to give a
different characterization of distinguished pairs.
Keywords:valued fields, non-Archimedean valued fields, irreducible polynomials Categories:12J10, 12J25, 12E05 |
4. CMB 2014 (vol 57 pp. 538)
Infinite Families of $A_4$-Sextic Polynomials In this article we develop a test to determine whether a sextic
polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic
to the alternating group $A_4$. This test does not involve the
computation of resolvents, and we use this test to construct several
infinite families of such polynomials.
Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomial Categories:12F10, 12F12, 11R32, 11R09 |
5. CMB 2014 (vol 57 pp. 735)
On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras |
On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras We describe of all finite
dimensional uniserial representations of a commutative associative
(resp. abelian Lie) algebra over a perfect (resp. sufficiently
large perfect) field. In the Lie case the size of the field
depends on the answer to following question, considered and solved
in this paper. Let $K/F$ be a finite separable field extension
and
let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some
non-zero elements $\alpha,\beta\in F$?
Keywords:uniserial module, Lie algebra, associative algebra, primitive element Categories:17B10, 13C05, 12F10, 12E20 |
6. CMB 2011 (vol 55 pp. 233)
On Algebraically Maximal Valued Fields and Defectless Extensions Let $v$ be a Henselian Krull valuation of a field $K$. In this paper,
the authors give some necessary and sufficient conditions for a
finite simple extension of $(K,v)$ to be defectless. Various
characterizations of algebraically maximal valued fields are also
given which lead to a new proof of a result proved by Yu. L. Ershov.
Keywords:valued fields, non-Archimedean valued fields Categories:12J10, 12J25 |
7. CMB 2011 (vol 55 pp. 821)
New Examples of Non-Archimedean Banach Spaces and Applications The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical non-archimedean feature. We determine whether these
extensions are of countable type, have $t$-orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm,
without non-zero vectors that are $\|\cdot\|$-orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$
into a valued field.
Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases Categories:46S10, 12J25 |
8. CMB 2007 (vol 50 pp. 588)
Cohomological Dimension and Schreier's Formula in Galois Cohomology Let $p$ be a prime and $F$ a field containing a primitive $p$-th
root of unity. Then for $n\in \N$, the cohomological dimension
of the maximal pro-$p$-quotient $G$ of the absolute Galois group
of $F$ is at most $n$ if and only if the corestriction maps
$H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open
subgroups $H$ of index $p$. Using this result, we generalize
Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp}
H^n(H,\Fp)$.
Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups Categories:12G05, 12G10 |
9. CMB 2007 (vol 50 pp. 313)
On Cauchy--Liouville--Mirimanoff Polynomials Let $p$ be a prime greater than or equal to 17 and
congruent to
2 modulo 3. We use results of Beukers and Helou on
Cauchy--Liouville--Mirimanoff
polynomials to show that
the intersection of the Fermat curve of degree $p$ with the
line $X+Y=Z$ in the projective plane
contains no algebraic points of degree
$d$ with $3 \leq d \leq 11$.
We prove a result on
the roots of these polynomials and show that, experimentally,
they seem to satisfy
the conditions of a mild extension of
an irreducibility theorem of P\'{o}lya and Szeg\"{o}.
These conditions are \emph{conjecturally}
also necessary for irreducibility.
Categories:11G30, 11R09, 12D05, 12E10 |
10. CMB 2006 (vol 49 pp. 11)
Going-Down Results for $C_{i}$-Fields We search for theorems that, given a $C_i$-field $K$ and a subfield $k$ of $K$, allow
us to conclude that $k$ is a $C_j$-field for some $j$. We give appropriate theorems in
the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$
is an algebraic extension. Here we are able to prove some results, and make conjectures. We
also point out the connection between these questions and Lang's conjecture on nonreal function
fields over a real closed field.
Keywords:$C_i$-fields, Lang's Conjecture Categories:12F, 14G |
11. CMB 2006 (vol 49 pp. 113)
$\PSL(2,2^n)$-Extensions Over $\mathbb F_{2^n}$ We construct a one-parameter generic polynomial for
$\PSL(2,2^n)$ over $\mathbb F_{2^n}$.
Categories:12F12, 12E10 |
12. CMB 2005 (vol 48 pp. 428)
Reduction of Elliptic Curves in Equal Characteristic~3 (and~2) and fibre type for elliptic curves
over discrete valued fields of equal characteristic~3.
Along the same lines, partial results are obtained
in equal characteristic~2.
Categories:14H52, 14K15, 11G07, 11G05, 12J10 |
13. CMB 2002 (vol 45 pp. 422)
On the Essential Dimension of Some Semi-Direct Products We give an upper bound on the essential dimension of the group
$\mathbb{Z}/q\rtimes(\mathbb{Z}/q)^*$ over the rational numbers,
when $q$ is a prime power.
Category:12F10 |
14. CMB 2002 (vol 45 pp. 388)
AlgÃ¨bres simples centrales de degrÃ© 5 et $E_8$ As a consequence of a theorem of Rost-Springer, we establish that the
cyclicity problem for central simple algebra of degree~5 on fields
containg a fifth root of unity is equivalent to the study of
anisotropic elements of order 5 in the split group of type~$E_8$.
Keywords:algÃ¨bres simples centrales, cohomologie galoisienne Categories:16S35, 12G05, 20G15 |
15. CMB 2002 (vol 45 pp. 71)
Images of Additive Polynomials in $\FF_q ((t))$ Have the Optimal Approximation Property We show that the set of values of an additive polynomial in several
variables with arguments in a formal Laurent series field over a
finite field has the optimal approximation property: every element in
the field has a (not necessarily unique) closest approximation in this
set of values. The approximation is with respect to the canonical
valuation on the field. This property is elementary in the language
of valued rings.
Categories:12J10, 12L12, 03C60 |
16. CMB 2001 (vol 44 pp. 313)
Images of mod $p$ Galois Representations Associated to Elliptic Curves We give an explicit recipe for the determination of the images
associated to the Galois action on $p$-torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.
Keywords:Galois groups, elliptic curves, Galois representation, isogeny Categories:11R32, 11G05, 12F10, 14K02 |
17. CMB 2001 (vol 44 pp. 223)
Extending the Archimedean Positivstellensatz to the Non-Compact Case A generalization of Schm\"udgen's Positivstellensatz is given which holds
for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not).
The proof is an extension of W\"ormann's proof.
Categories:12D15, 14P10, 44A60 |
18. CMB 1999 (vol 42 pp. 354)
A Real Holomorphy Ring without the SchmÃ¼dgen Property A preordering $T$ is constructed in the polynomial ring $A = \R
[t_1,t_2, \dots]$ (countably many variables) with the following two
properties: (1)~~For each $f \in A$ there exists an integer $N$
such that $-N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$.
(2)~~For all $f \in A$, if $N+f, N-f \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgen-W\"ormann result that for any preordering $T$ in a
finitely generated $\R$-algebra $A$, if property~(1) holds, then
for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$.
Also, adjoining to $A$ the square roots of the generators of $T$
yields a larger ring $C$ with these same two properties but with
$\Sigma C^2$ (the set of sums of squares) as the preordering.
Categories:12D15, 14P10, 44A60 |
19. CMB 1998 (vol 41 pp. 214)
On a problem of Rubel concerning the set of functions satisfying all the algebraic differential equations satisfied by a given function |
On a problem of Rubel concerning the set of functions satisfying all the algebraic differential equations satisfied by a given function For two functions $f$ and $g$, define $g\ll f$ to mean that $g$ satisfies
every algebraic differential equation over the constants satisfied by $f$.
The order $\ll$ was introduced in one of a set of problems on algebraic
differential equations given by the late Lee Rubel. Here we characterise
the set of $g$ such that $g\ll f$, when $f$ is a given Liouvillian function.
Categories:34A34, 12H05 |
20. CMB 1998 (vol 41 pp. 3)
Root closure in Integral Domains, III {If A is a subring of a commutative ring B and if n
is a positive integer, a number of sufficient conditions are given for
``A[[X]]is n-root closed in B[[X]]'' to be equivalent to ``A is n-root
closed in B.'' In addition, it is shown that if S is a multiplicative
submonoid of the positive integers ${\bbd P}$ which is generated by
primes, then there exists a one-dimensional quasilocal integral domain
A (resp., a von Neumann regular ring A) such that $S = \{ n \in {\bbd P}\mid
A$ is $n$-root closed$\}$ (resp., $S = \{n \in {\bbd P}\mid A[[X]]$
is $n$-rootclosed$\}$).
Categories:13G05, 13F25, 13C15, 13F45, 13B99, 12D99 |
21. CMB 1997 (vol 40 pp. 81)
Une caractÃ©risation des corps satisfaisant le thÃ©orÃ¨me de l'axe principal Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\`eme
de l'axe principal \`a l'aide de propri\'et\'es des formes
carac\-t\'erisation de ces m\^emes corps due \`a Waterhouse,
on retrouve \`a partir de l\`a, de fa\c{c}on \'el\'ementaire,
un r\'esultat de Becker selon lequel un pro-$2$-groupe qui se
r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est
engendr\'e par des involutions.
ABSTRACT. We characterize general fields $K$, satisfying the
Principal Axis Theorem, by means of properties of trace forms of
the finite extensions of $K$. From this and Waterhouse's
characterization of the same fields, we rediscover, in quite an
elementary way, a result of Becker according to which a
pro-$2$-group which occurs as the absolute Galois group of such
a field $K$, is generated by
Categories:11E10, 12D15 |