Concerning the Feit-Thompson conjecture, Stephens found an example for primes 17 and 3313. In this paper, using the Artin map, these primes are common index divisors of subfields of a cyclotomic field, and some results in [7,8] shall be again proved.

We give a reduction theorem for Brauer’s height zero conjecture.

In this paper, we give a basis for the derivation module of the cone over the Shi arrangement of the type $D_{\ell}$ explicitly.

We show that the equation (1) in the text defines a multiplicative excellent family of elliptic surfaces (or of cubic surfaces) with Galois group isomorphic to the Weyl group of type $E_{6}$. The main properties of the family are formulated as Theorems 1 and 2 in § 3.