18,831 research outputs found
Further results on the Morgan-Mullen conjecture
Let be the finite field of characteristic with
elements and its extension of degree . The conjecture of
Morgan and Mullen asserts the existence of primitive and completely normal
elements (PCN elements) for the extension for
any and . It is known that the conjecture holds for . In this
work we prove the conjecture for a larger range of exponents. In particular, we
give sharper bounds for the number of completely normal elements and use them
to prove asymptotic and effective existence results for , where for the asymptotic results and
for the effective ones. For even we need to assume that
.Comment: arXiv admin note: text overlap with arXiv:1709.0314
A construction of polynomials with squarefree discriminants
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we
give an unconditional construction of infinitely many monic irreducible
polynomials of degree n with integer coefficients having squarefree
discriminant and exactly r real roots. These give rise to number fields of
degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we
may also force the discriminant to be coprime to any given integer. The number
of fields produced with discriminant in the range [-N, N] is at least c
N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic
number fields admit everywhere unramified degree n extensions whose normal
closures have Galois group A_n. This generalizes results of Yamamura, who
treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not
control the real place.Comment: 10 pages; v2: refereed version, minor edits onl
Practical improvements to class group and regulator computation of real quadratic fields
We present improvements to the index-calculus algorithm for the computation
of the ideal class group and regulator of a real quadratic field. Our
improvements consist of applying the double large prime strategy, an improved
structured Gaussian elimination strategy, and the use of Bernstein's batch
smoothness algorithm. We achieve a significant speed-up and are able to compute
the ideal class group structure and the regulator corresponding to a number
field with a 110-decimal digit discriminant
Sieving rational points on varieties
A sieve for rational points on suitable varieties is developed, together with
applications to counting rational points in thin sets, the number of varieties
in a family which are everywhere locally soluble, and to the notion of friable
rational points with respect to divisors. In the special case of quadrics,
sharper estimates are obtained by developing a version of the Selberg sieve for
rational points.Comment: 30 pages; minor edits (final version
The cyclic sieving phenomenon: a survey
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a
2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and
f(q) be a polynomial in q with nonnegative integer coefficients. Then the
triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we
have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g,
and w is a root of unity chosen to have the same order as g. It might seem
improbable that substituting a root of unity into a polynomial with integer
coefficients would have an enumerative meaning. But many instances of the
cyclic sieving phenomenon have now been found. Furthermore, the proofs that
this phenomenon hold often involve interesting and sometimes deep results from
representation theory. We will survey the current literature on cyclic sieving,
providing the necessary background about representations, Coxeter groups, and
other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes
suggested by colleagues and the referee. To appear in the London Mathematical
Society Lecture Note Series. The third version has a few smaller change
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