  In the past two decades, many researchers have studied {\it index $2$} Gauss
sums, where the group generated by the characteristic $p$ of the underling
finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$
for the order $m$ of the multiplicative character involved. A complete solution
to the problem of evaluating index $2$ Gauss sums was given by Yang and
Xia~(2010). In particular, it is known that some nonzero integral powers of the
Gauss sums in this case are in quadratic fields. On the other hand,
Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004, 2012)
studied {\it pure} Gauss sums, some nonzero integral powers of which are in the
field of rational numbers. In this paper, we study Gauss sums, some integral
powers of which are in quadratic fields. This class of Gauss sums is a
generalization of index $2$ Gauss sums and an extension of pure Gauss sums to
quadratic fields.
