Let $p$ be a prime number, and $a$ $(\in \mathbf{Q}^{\times})$ a rational number. Then, F. Kawamoto proved that the cyclic extension $\mathbf{Q}(\zeta_p, a^{1/p})/\mathbf{Q}(\zeta_p)$ has a normal integral basis if it is at most tamely ramified. We give some generalized version of this result replacing the base field $\mathbf{Q}$ with some real abelian fields of prime power conductor.

T. Ono raised some problems on relations between Ono's numbers $p_D$ and the class numbers $h_D$ of imaginary quadratic fields. In this paper we give an upper bound for $p_D$. The upper bound contributes to one of the problems.

We show the validity of the interface equation which describes the behavior of the support of the solution to the one-dimensional porous media equation with absorption. Our proof is based on the finite difference methods.