Iwasawa invariants and class number parity of multi-quadratic number fields

In this paper, based mainly on the method of Iwasawa and Kida, by studying in detail the Hasse units and the ramifications of prime ideals, we obtain explicit results of Iwasawa invariants $\lambda_{2}$ of the cyclotomic $\Z_{2}-$extensions of number fields. In particular, under the Greenberg's conjecture, we obtain an explicit formula of $\lambda_{2}$ for imaginary multi-quadratic number fields. As an application, we give a criteria of determining class number parity of multi-quadratic number fields.