CLASS NUMBER PARITY OF A QUADRATIC TWIST OF A CYCLOTOMIC FIELD OF PRIME POWER CONDUCTOR

Let p be a fixed odd prime number and K-n the p(n+1)-st cyclotomic field. For a fixed integer d is an element of Z with root d is not an element of K-0, denote by L-n the imaginary quadratic subextension of the biquadratic extension K-n(root d)/K-n(+) with L-n not equal K-n. Let h(n)* and h(n)(-) be the relative class numbers of K-n and L-n, respectively. We give an explicit constant n(d) depending on p and d such that (i) for any integer n >= n(d), the ratio h(n)(-)/h(n-1)(-) is odd if and only if h(n)*/h(n-1)* is odd and (ii) for 1 <= n < n(d), h(n)(-)/h(n-1)(-) is even.