A dihedral number field is a non-normal quartic field K which possesses a quadratic subfield k. That is, for some integer α of k. Integral bases of these fields were known by Sommer (1907), but the form in which they were known was of little use for computational purposes. In this paper we construct integral bases of those dihedral fields with quadratic subfield of the form , d ≢ 1 (mod 8), for which only rational quantities need be determined. Although the general theory may easily be generalized to the case d ≡ 1 (mod 8), the actual determination of integral bases in this case is left to a later paper.