Let k be an algebraic number field of finite degree and be a prime ideal of k, lying above a rational prime p. We denote by G () the multiplicative group of residue classes modulo (N ≧ 0) which are relatively prime to . The structure of G () is well-known, when N = 0, or k is the rational number field Q. If k is a quadratic number field, then the direct decomposition of G () is determined by A. Ranum [6] and F.H-Koch [4] who gives a basis of a group of principal units in the local quadratic number field according to H. Hasse [2]. In [5, Theorem 6.2], W. Narkiewicz obtains necessary and sufficient conditions so that G () is cyclic, in connection with a group of units in the -adic completion of k.

It is interesting to know what kinds of primes are the factors of the class number of an algebraic number field, and especially to find ones being prime to the degree. About this matter it is desirable to construct the unramified Abelian extensions plainly. In this paper we shall show some of them for the prime cyclotomic number field and its quadratic extensions using the units of subfields.Let I be an odd prime and ζ be a primitive /-th root of unity. Let k = Q(ζ) be the /-th cyclotomic number field over the field Q of rationals. If / is irregular, then there is an even integer r with 2 1 of Q(

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The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

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