A sufficient condition for Leopoldtʹs conjecture

In this paper, by using an analogue of theorems of Iwasawa (Kenkichi Iwasawa Collected Papers, vol. 2, Springer, Berlin, 2001, pp. 862–870) we give a sufficient condition for Leopoldtʹs conjecture (J. Reine Angew. Math. 209 (1962) 54) on the non-vanishing of the p-adic regulator of an algebraic number field. Using this sufficient condition we are able to prove Leopoldtʹs conjecture for several non-Galois extensions over the rational number field .