In the paper Iwasawa *On Galois groups of local fields*, he proves that if $ V $ is the maximum slightly branched extension of $ mathbb {Q} _p $with the Galois group $ Gamma $ on its base, then its abelianized absolute group of Galois (that is, the abelianized wild inertia group of $ mathbb {Q} _p $) sits in a short, exact sequence

$$ 1 to R to G to mathbb {Z} _p (1) to 1 $$

or $ R $ is the regular representation of $ Gamma $ on the set of $ p $transitional measures on $ Gamma $and the action of conjugation on $ mathbb {Z} _p (1) $ is given by the usual cyclotomic character and then the identification of $ mathbb {Z} / (p-1) (1) $ with the $ (p-1) $roots of the unit of the tame group of inertia. (He actually proves a similar thing on any degree $ d $ extension of $ mathbb {Q} _p $, or $ R $ is replaced by $ R ^ d $)

Can we explicitly describe the extensions associated with any quotient of this group explicitly described? In particular, I ask this question because it looks like what the extension associated with the quotient $ mathbb {Z} _p (1) $ This should be obvious, but that's not it. the $ p $The cyclotomic extension, after all, should have the trivial conjugation action by cyclotomy.