In research of algebraic number fields, a construction of an appropriate integral basis plays a fundamental role. Researchers impose certain conditions on algebraic number fields in order to handle the hard problems of Algebraic Number Theory. For construction of an appropriate integral basis and determination of the relative monogenity and absolute monogenity, we select an algebraic number field, which is a composite field of cyclotomic field of conductor n and a totally real field of conductor m with (n;m) = 1 and also cyclic sextic field of prime conductor p with the prime discriminant p_: In this thesis we consider a classical problem of Algebraic Number Theory that an algebraic number field is monogenic or not, which was introduced in the 1960s by a German mathematician Helmut Hasse. In the case of composite field K = kn _ F; the methodology begins with the determination of units in the cyclotomic field kn to show that Zkn = Zk+ n [_]; where k+ n is the maximal real subfield of the cyclotomic field kn: By the consideration of any element of ZK taking the partial di_erent and its norm, we conclude that ZK has no power integral basis. Our methodology in the case of cyclic sextic field L begins with an algebraic integer _0 of L; where _0 denotes the Gau_ period of length p?1 6 : We established the non monogenic phenomenon in L by taking the relative norm NL=k(_0?__ 0 );NL=k(_0?__2 0 ) and NL=k(_0 ?__3 0 ) of the three partial factors _0 ?__ 0 ; _0 ?__2 0 and _0 ?__3 0 respectively of the di_erent dL(_0) by the way of the quadratic subfield k of L: Here _ is an automorphism _p ! _r p ; where r a primitive root modulo p and _p is a primitive pth root of unity. We conclude that _0 generates the power integral basis for the 7th cyclotomic field, maximal real subfield of 13th cyclotomic field and a field of conductor 32 only. In fact for any element _ of L; we have shown that _ cannot generate a power integral basis in the same way as _0 except for the above three sextic fields.