On The Monogenity of Cyclic Sextic Fields of Prime Conductor

Abstract

In research of algebraic number elds, a construction of an appropriate integral basis plays a fundamental role. Researchers impose certain conditions on algebraic number elds 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 eld, which is a composite eld of cyclotomic eld of conductor n and a totally real eld of conductor m with (n;m) = 1 and also cyclic sextic eld 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 eld is monogenic or not, which was introduced in the 1960s by a German mathematician Helmut Hasse. In the case of composite eld K = kn F; the methodology begins with the determination of units in the cyclotomic eld kn to show that Zkn = Zk+ n [ ]; where k+ n is the maximal real sub eld of the cyclotomic eld 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 eld 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 sub eld 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 eld, maximal real sub eld of 13th cyclotomic eld and a eld 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 elds.