Bicyclic biquadratic fields which contain irreducible rational primes (Q5939142)

A rational prime may be irreducible but still not generate a prime ideal in an algebraic number field. The authors intend to characterize the normal extensions of the rational numbers which contain such rational primes that split completely, but are irreducible, and call such primes ``sci primes''. They first show that a normal extension of the rationals of prime degree \(\ell=2,3\) or 5 contains sci primes if and only if its class number is greater than 1, and moreover that any number field of degree greater than the Davenport constant of its class group does not contain sci primes. Next, they consider bicyclic, biquadratic fields and give sufficient conditions for such fields to contain sci primes. Finally, they obtain some precise conditions for imaginary bicyclic biquadratic fields to contain sci primes.