Normal bases and irreducible polynomials

Let $\mathbb{F}_q$ denote the finite field of $q$ elements and $\mathbb{F}_{q^n}$ the degree $n$ extension of $\mathbb{F}_q$. A normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F} _q$ is a basis of the form $\{\alpha,\alpha^q,\dots,\alpha^{q^{n-1}}\}$. An irreducible polynomial in $\mathbb{F} _q[x]$ is called an $N$-polynomial if its roots are linearly independent over $\mathbb{F} _q$. Let $p$ be the characteristic of $\mathbb{F} _q$. Pelis et al. showed that every monic irreducible polynomial with degree $n$ and nonzero trace is an $N$-polynomial provided that $n$ is either a power of $p$ or a prime different from $p$ and $q$ is a primitive root modulo $n$. Chang et al. proved that the converse is also true. By comparing the number of $N$-polynomials with that of irreducible polynomials with nonzero traces, we present an alternative treatment to this problem and show that all the results mentioned above can be easily deduced from our main theorem.Comment: This is my first submission to arxiv. Just a try

Rare decay B→Xsl+l−B\to X_sl^+l^- in a CP spontaneously broken two Higgs doublet model

The Higgs boson mass spectrum and couplings of neutral Higgs bosons to fermions are worked out i n a CP spontaneously broken two Higgs doublet model in the large tan$\beta$ case. The differential branching ratio, forward-backward asymmetry, CP asymmetry and lepton polarization for $B\to X_s l^+ l^-$ are computed. It is shown that effects of neutral Higgs bosons are quite significant when $\tan\beta$ is large. Especially, the CP violating normal polarization $P_N$ can be as large as several percents.Comment: 27 pages, text updated, new numerical results include

A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems

By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that when α is big enough, it has an eigenvalue at 1 with multiplicity at least n, and the remaining eigenvalues are all located in a unit circle centered at 1. Particularly, when the preconditioner is used in general saddle point problems, it guarantees eigenvalue at 1 with the same multiplicity, and the remaining eigenvalues will tend to 1 as the parameter α→0. Consequently, this can lead to a good convergence when some GMRES iterative methods are used in Krylov subspace. Numerical results of Stokes problems and Oseen problems are presented to illustrate the behavior of the preconditioner