Let $x^{p^k}-a$ be irreducible over $\mathbf{Q}$. The first part of this paper is to explicitly give the decomposition rules for the factorization of $p$ in the ring of integers of $\mathbf{Q}(a^{1/p^k})$.

As an application of the above we use these results to determine the genus field of $\mathbf{Q}(a^{1/n})$, where $x^n-a$ is irreducible over $\mathbf{Q}$ and we make no restrictions on $a$.

A weighted endomorphism of an algebra is an endomorphism followed by a multiplier. In [6] and [4], H. Kamowitz characterized compact weighted endomorphisms of $C(X)$ and the disc algebra. In this note we define a weighted composition operator on a function algebra as a generalization of a weighted endomorphism, and characterize compact weighted composition operators on a function algebra satisfying a certain condition [Theorem 2]. This theorem not only includes Kamowitz's results as corollaries, but also has an application to compact weighted composition operators on the Hardy class $H^\infty(D)$.

In this note we give some interesting topological restrictions for the immersion of Kaehler surfaces into the complex projective space $CP^m(1)$.