Let ?1 and ?2 be primes such that ?1≡?2≡5(mod8) , ?=−1‾‾‾√ , ?=2?1?2 , ?=ℚ(?‾‾√,?) , ?(1)2 be the Hilbert 2-class field of ? , ?(2)2 be the Hilbert 2-class field of ?(1)2 , ? be the Galois group of ?(2)2/? and ?(∗)=ℚ(?1‾‾√,?2‾‾√,2‾√,?) be the genus field of ? . The 2-part ??,2 of the class group of ? is of type (2,2,2) . Our goal is to study the 2-class field tower of ? and to calculate the order of ? .
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