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Yasuhiro Kishi

Publications and source records attributed to Yasuhiro Kishi.

3 recordsLinked to original sources
arXiv2019

5-Class towers of cyclic quartic fields arising from quintic reflection

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The precise length ell5(M) of the 5-class tower of M is determined, when G(5,2)M is of order less than or equal to 5^5. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0<d<10^4, respectively -2*10^5<d<0.

math.NT↗
arXiv2018

A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$

We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb Q(\sqrt{p})$, certain units which are roots of a parametric quartic polynomial, the Kummer theory, the Gauss sums and the Jacobi sums, linear recurrence sequences, a consequence of the Weil conjecture and a result of Lenstra which is a generalization of Artin conjecture on primitive roots. Our result is based on the famous Scholz' results on pairs of quadratic fields $\mathbb Q(\sqrt{D})$ and $\mathbb Q (\sqrt{-3D})$.

math.NT↗