  Investigations of monogenity and power integral bases were recently extended
from the absolute case (over Q) to the relative case (over algebraic number
fields). Formerly, in the relative case we only succeeded to calculate
generators of power integral bases when the ground field is an imaginary
quadratic field. This is the first case when we consider monogenity in the more
difficult case, in extensions of real quadratic fields. We give efficient
algorithms for calculating generators of power integral bases in cubic and
quartic extensions of real quadratic fields, more exactly in composites of
cubic and quartic fields with real quadratic fields. In case the quartic field
is totally complex, we present an especially simple algorithm. We illustrate
our method with two detailed examples.
