65 research outputs found
Irreducible values of polynomials
Schinzel's Hypothesis H is a general conjecture in number theory on prime
values of polynomials that generalizes, e.g., the twin prime conjecture and
Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic
analog of this conjecture for polynomial rings over pseudo algebraically closed
fields. This implies results over large finite fields. A main tool in the proof
is an irreducibility theorems \`a la Hilbert.Comment: The paper was shorten, some errors were correcte
Hardy-Littlewood tuple conjecture over large finite field
We prove the following function field analog of the Hardy-Littlewood
conjecture (which generalizes the twin prime conjecture) over large finite
fields. Let n,r be positive integers and q an odd prime power. For distinct
polynomials a_1, ..., a_r over F_q of degree <n let \pi(q,n;a) be the number of
monic polynomials f over F_q of degree n such that f+a_1, ..., f+a_r are
simultaneously irreducible. We prove that \pi(q,n;a) asymptotically equals
q^n/n^r as q tends to infinity on odd prime powers and n,r are fixed (the tuple
a1,...,a_r need not be fixed).Comment: minor change
Projective pairs of profinite groups
We generalize the notion of a projective profinite group to a projective pair
of a profinite group and a closed subgroup. We establish the connection with
Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an
algebraic extension of a PAC field K. Then M/K is PAC if and only if the
corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective.
Moreover any projective pair can be realized as absolute Galois groups of a PAC
extension of a PAC field.
Using this characterization we construct new examples of PAC extensions of
relatively small fields, e.g., unbounded abelian extensions of the rational
numbers
Irreducibility and embedding problems
We study irreducible specializations, in particular when group-preserving
specializations may not exist. We obtain a criterion in terms of embedding
problems. We include several applications to analogs of Schinzel's hypothesis H
and to the theory of Hilbertian fields
On the Bateman-Horm Conjecture about Polynomial Rings
Given a power of a prime number and "nice" polynomials
f_1,...,f_r\in\bbF_q[T,X] with if , we establish an asymptotic
formula for the number of pairs (a_1,a_2)\in\bbF_q^2 such that
are irreducible in \bbF_q[T]. In
particular that number tends to infinity with .Comment: To be published in Muenster Journal of Mathematic
On totally ramified extensions of discrete valued fields
We give a simple characterization of the totally wild ramified valuations in
a Galois extension of fields of characteristic p. This criterion involves the
valuations of Artin-Schreier cosets of the F_{p^r}^\times-translation of a
single element. We apply the criterion to construct some interesting examples
On the number of ramified primes in specializations of function fields over
We study the number of ramified prime numbers in finite Galois extensions of
obtained by specializing a finite Galois extension of
. Our main result is a central limit theorem for this number. We
also give some Galois theoretical applications.Comment: To appear in the New York Journal of Mathematic
PAC Fields over Finitely Generated Fields
We prove the following theorem for a finitely generated field : Let be
a Galois extension of which is not separably closed. Then is not PAC
over .Comment: 7 pages, Math.
Subgroup structure of fundamental groups in positive characteristic
Let be the \'etale fundamental group of a smooth affine curve over an
algebraically closed field of characteristic . We establish a criterion
for profinite freeness of closed subgroups of . Roughly speaking, if a
closed subgroup of is "captured" between two normal subgroups, then it is
free, provided it contains most of the open subgroups of index . In the
proof we establish a strong version of "almost -freeness" of and
then apply the Haran-Shapiro induction
Irreducible polynomials of bounded height
The goal of this paper is to prove that a random polynomial with i.i.d.
random coefficients taking values uniformly in is
irreducible with probability tending to as the degree tends to infinity.
Moreover, we prove that the Galois group of the random polynomial contains the
alternating group, again with probability tending to .Comment: Some revision
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