The problem of explicitly
constructing classfields (Hilbert’s Twelfth problem) is largely unresolved, except when
a classfield is absolutely abelian or abelian over an imaginary quadratic number field.
Here an explicit construction of certain split extensions of number fields is
given, which maintains control over the primes which ramify. This naturally
leads to the construction of cyclic classifields over a given number field. An
algorithm is provided to obtain the minimal polynomials for the generating
elements of the extension constructed. The methods employed here rely heavily
on classfield theory and the properties of Lagrange resolvents and group
determinants.
