文摘
We disprove, by means of numerical examples and theoretical arguments, illustrated with <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=3722af86593da037dd61cda4230271cb" title="Click to view the MathML source">p=3span><span class="mathContainer hidden"><span class="mathCode">span>span>span>, the existence of a Riemann–Hurwitz formula for the p-ranks of relative class groups in a p-ramified p -extension <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=78adbca67146b8b3320f88286ea0cfca" title="Click to view the MathML source">K/kspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> of number fields of CM-type containing <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=3bc73f80f273a1fe2b76aad053fa0fc8" title="Click to view the MathML source">μ<sub>psub>span><span class="mathContainer hidden"><span class="mathCode">span>span>span> in contradiction with a result published in 1996. In the cyclic case of degree p, under some assumptions on the p-class group of k and the decomposition of the p-places, we prove some results on the structure of the p-class group of K and justify that some theoretical structures do not exist in this particular situation. In this context, an analogue of Kida's formula is valid for the p-ranks if and only if the p-class group of K is reduced to the group of ambiguous classes, which is not always the case, as shown by a numerical table for <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301998&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301998&_rdoc=1&_issn=0022314X&md5=3722af86593da037dd61cda4230271cb" title="Click to view the MathML source">p=3span><span class="mathContainer hidden"><span class="mathCode">span>span>span>.