Arbitrarily large Pólya groups of compositum of simplest fields and some bi-quadratic Pólya fields

The Pólya group [Formula: see text] of an algebraic number field [Formula: see text] is the subgroup of the ideal class group [Formula: see text] generated by the ideal classes of the products of prime ideals of the same norm. If [Formula: see text] is trivial, then the number field [Formula: see text] is said to be a Pólya field. In the first part of this paper, we extend a recent result of ours and prove the existence of infinitely many number fields having Pólya groups as large as possible and those number fields being the compositum of a simplest cubic field and a quintic field. Using a result of Zylinski, we also prove that those fields are of index [Formula: see text]. Next, we produce three new families [Formula: see text], [Formula: see text] and [Formula: see text] of totally real bi-quadratic Pólya fields [Formula: see text] involving prime numbers [Formula: see text] and [Formula: see text] that satisfy certain quadratic residue conditions among themselves. It is worthwhile to note that in each of these fields, exactly five primes ramify in [Formula: see text] and this is the maximum possible number of ramified primes in a bi-quadratic Pólya field over [Formula: see text].