Some numerical invariants related to central embedding problems (Q1892567)

For central embedding problems of fields with prescribed ramified primes and with kernel group cyclic of order a power of a prime \(p\) three numerical invariants are defined in cohomological terms. They are equal to 1 if the embedding problem is solvable. The following result is proved. If one lets the embedding problem vary such that only two things are fixed -- the base field \(k\), a number field for which Leopoldt's conjecture holds for \(p\), and the set of prescribed ramified primes, then these three invariants cannot become arbitrarily high.