【摘 要】
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Let q ≥ 5 be a prime number.Let be a quadratic number field,where d = Then the class number of k is divisible by q for certain integers u,w.Conversely,assume
【机 构】
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Department of Mathematical Sciences
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Let q ≥ 5 be a prime number.Let be a quadratic number field,where d = Then the class number of k is divisible by q for certain integers u,w.Conversely,assume Ω/ k is an unramified cyclic extension of degree q Ω (which implies the class number of k is divisible by q),and Ω is the splitting field of some irreducible trinomial f(X) = Xq-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) = Xq-uq-2wX-uq-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-1uwq-1uw-+u2qq).Moreover,the above two results are both generalized for certain kinds of general polynomials.
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