Quadratic Number Fields with Class Numbers Divisible by a Prime q

Let q≥5 be a prime number. Let k=Q(√d) be a quadratic number field, where d=(-1)q(q-1)/2. (-(q-1)^q-1UW^q u^2q^q). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume Ω/k is an unramifled cyclic extension of degree q (which implies the class number of k is divisible by q), and Ω2 is the splitting field of some irreducible trinomial f(X)=X^q-aX-b with integer coefficients, k=Q(√D(f)) with D(f) the discriminant of f(X). Then f(X) must be of the form f(X)=X^q-u^q-2 wX-u^q-1 in a certain sense where u,w are certain integers. Therefore, k=Q(√d) with d=(-1)^q(q-1)/2 (-(q-1)^q-1uw^q u^2q^q). Moreover, the above two results are both generalized for certain kinds of general polynomials.

Quadratic Number Fields with Class Numbers Divisible by a Prime q

Let q  5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1) ). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1) ). Moreover, the above two results are both generalized for certain kinds of general polynomials.