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 carpetpool 2018-01-30 06:10

Cyclic fields with class number h

Is it true that for any prime p = 1 (mod n), there is a polynomial P of degree n generating a cyclic field K = Q(ζ) where ζ is a root of P and K has class number p?

This is necessary to study certain structures mod p. For example, p = 13, we have the vector of exponents F0=[0], F1=[1, 5, 8, 12], F2=[3, 2, 11, 10], F3=[9, 6, 7, 4]. Each of these vectors Fn represents an ideal class group. F2 and F3 are cosets of F1, and F0 is the zero group, or principal ideal class group. To find an example, one would need to find a cyclic quartic field with class number p = 13. I don't know of such an example, is someone able to give a generating polynomial for such a field?

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