CONSTRUCTION OF MAXIMAL UNRAMIFIED p-EXTENSIONS WITH PRESCRIBED GALOIS GROUPS

For any number field F (not necessary of finite degree) and prime number p, let Lp(F ) denote the maximal unramified p-extension over F , and put G̃F (p) = Gal(Lp(F )/F ). Though the structure of G̃F (p) has been one of the most fascinating theme of number theory, our knowledge on it is not enough even at present: It had been a cerebrated open problem for a long time whether G̃F (p) can be infinite for a number field F of finite degree, and Golod and Shafarevich solved it b y giving F with infinite G̃F (p). Hence we barely know that G̃F (p) can be infinite. However we do not know exactly what kind of pro-p-groups occur as G̃F (p); for example, there are no examples of infinite G̃F (p) for number fields F of finite degree whose structure is completely determined. The known general property of the group G̃F (p) for number fields F of finite degree is only that G̃F (p) is a finitely presented pro-p-group any whose open subgroup has finite abelianization, which comes from rather fundamental facts of algebraic number theory, namely, class field theory and the finiteness of the ideal class group. Indeed, a consequence of the Fontaine-Mazur conjecture predicts that G̃F (p) has a certain distinguished property (see [2]), however, this conjecture seems far reach object at present. On the other hand, we have known that various kind of pro-p-groups, especially, finite p-groups in fact occur as G̃F (p). For example, Scholz and Taussky [11] have already determined the structure of G̃F (p) for F = Q( √ −4027) and p = 3 in 1930’s : this group is a non-abelian finite group of order 3. Also, Yahagi [13] showed that for any given finite abelian p-group A there exists an number field F of finite degree such that G̃F (p) ab ≃ A.