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

• Published in: The Ramanujan journal Vol. 52; no. 1; pp. 133 - 161
• Main Authors: Aoki, Miho, Kishi, Yasuhiro
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2020; Springer Nature B.V
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Integers; Mathematics; Mathematics and Statistics; Number Theory; Polynomials; 11R29; Abelian number fields; Gauss sums; Class numbers; Fundamental units; 11R11; Linear recurrence sequences; 11R16; Jacobi sums

Let p be a prime number with p ≡ 5 ( mod 8 ) . We construct a new infinite family of pairs of imaginary cyclic fields of degree ( p - 1 ) / 2 with both class numbers divisible by p . Let k 0 be the unique subfield of Q ( ζ p ) of degree ( p - 1 ) / 4 and u p = ( t + b p ) / 2 ( > 1 ) be the funda...