1,283 research outputs found
The Grunwald problem and specialization of families of regular Galois extensions
We investigate specializations of infinite families of regular Galois
extensions over number fields. The problem to what extent the local behaviour
of specializations of one single regular Galois extension can be prescribed has
been investigated by D\`ebes and Ghazi in the unramified case, and by Legrand,
Neftin and the author in general. Here, we generalize these results and give a
partial solution to Grunwald problems using Galois extensions arising as
specializations of a family of regular Galois extensions.Comment: 21
On rational functions with monodromy group
We compute new polynomials with Galois group over .
These polynomials stem from various families of covers of
ramified over at least 4 points. Each of these
families has features that make a detailed study interesting. Some of the
polynomials lead, via specialization, to number fields with very small
discriminant or few ramified primes.Comment: 11 page
Solvability of Generalized Monomial Groups
The solvability of monomial groups is a well-known result in character
theory. Certain properties of Artin L-series suggest a generalization of these
groups, namely to such groups where every irreducible character has some
multiple which is induced from a character phi of U with solvable factor group
U/ker(phi). Using the classification of finite simple groups, we prove that
these groups are also solvable. This means in particular that the mentioned
properties do not enable one to deduce a proof of the famous Artin conjecture
for any non-solvable group from a possible proof for solvable groups.Comment: 21 page
On intersective polynomials with non-solvable Galois group
We present new theoretical results on the existence of intersective
polynomials with certain prescribed Galois groups, namely the projective and
affine linear groups and as well as the affine
symplectic groups . For
further families of affine groups, existence results are proven conditional on
the existence on certain tamely ramified Galois extensions. We also compute
explicit families of intersective polynomials for certain non-solvable groups.Comment: 16 page
Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups
We compute the first explicit polynomials with Galois groups , , and over .
Furthermore we compute the first examples of totally real polynomials with
Galois groups , , and over
. All these examples make use of families of covers of the
projective line ramified over four or more points, and therefore use techniques
of explicit computations of Hurwitz spaces. Similar techniques were used
previously e.g. by Malle, Couveignes, Granboulan and Hallouin. Unlike previous
examples, however, some of our computations show the existence of rational
points on Hurwitz spaces that would not have been obvious from theoretical
arguments
A note on the product of two permutations of prescribed orders
We prove a conjecture by Stefan Kohl on the existence of triples of
permutations of bounded degree with prescribed orders and product 1. This
result leads to an existence result for covers of the complex projective line
with bounded degree and prescribed ramification indices.Comment: 8 page
On the number of cyclic transitive subgroups of a permutation group
We prove an upper bound for the number of cyclic transitive subgroups in a
finite permutation group and clarify the structure of the groups for which this
bound becomes sharp. We also give an application in the theory of number
fields.Comment: 8 page
Density results for specialization sets of Galois covers
We provide evidence for this conclusion: given a finite Galois cover of group , almost all (in a density
sense) realizations of over do not occur as specializations of
. We show that this holds if the number of branch points of is
sufficiently large, under the abc-conjecture and, possibly, the lower bound
predicted by the Malle conjecture for the number of Galois extensions of
of given group and bounded discriminant. This widely extends a
result of Granville on the lack of -rational points on quadratic
twists of hyperelliptic curves over with large genus, under the
abc-conjecture (a diophantine reformulation of the case
of our result). As a further evidence, we exhibit a
few finite groups for which the above conclusion holds unconditionally for
almost all covers of of group . We also introduce
a local-global principle for specializations of Galois covers and show that it often fails if has abelian Galois
group and sufficiently many branch points, under the abc-conjecture. On the one
hand, such a local-global conclusion underscores the "smallness" of the
specialization set of a Galois cover of . On the other
hand, it allows to generate conditionally "many" curves over
failing the Hasse principle, thus generalizing a recent result of Clark and
Watson devoted to the hyperelliptic case.Comment: 37 page
Some examples of quadratic fields with finite nonsolvable maximal unramified extensions II
Let be a number field and be the maximal extension of that
is unramified at all places. In a previous article, the first author found
three real quadratic fields such that is finite and
nonabelian simple under the assumption of the GRH(Generalized Riemann
Hypothesis). In this article, we will identify more quadratic number fields
such that is a finite nonsolvable group and also explicitly
calculate their Galois groups under the assumption of the Generalized Riemann
Hypothesis.Comment: 22 page
Unramified extensions over low degree number fields
For various nonsolvable groups , we prove the existence of extensions of
the rationals with Galois group and inertia groups of order
dividing , where is the smallest exponent of a generating set
for . For these groups , this gives the existence of number fields of
degree with an unramified -extension. The existence of such
extensions over for all finite groups would imply that, for every
finite group , there exists a quadratic number field admitting an unramified
-extension, as was recently conjectured. We also provide further evidence
for the existence of such extensions for all finite groups, by proving their
existence when is replaced with a function field where
is an ample field
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