Some Power Integral Bases Of Q(ζ₁₅)

In this thesis, we discuss the generators of power bases of some number fields. First we study the maximal real subfield Q(ζ₁₅ + ζ₁₅^-1) of Q(ζ₁₅) which is a quartic field. We can get that the power base exist since the ring of integers is Z[ζ₁₅ + ζ₁₅^-1]. The set of generators is stable under integer translation since Z[α] = Z[n + α). As an application of index form equations in quartic number field, we get all generators of power bases. In chapter 2, our goal is to extend the basic ideas of that method of determining generator of power integral bases to relative quartic extension field Q(ζ₁₅) over Q(ζ₃). Relative Thue equation play an essential role in solving all generators of relative power bases. We get six form generators by means of the solutions of some relative Thue equations. In chapter 3, we restrict our attention to Q(ζ₁₅). It is well-know that Z[ζ₁₅] is the ring of integers of Q(ζ₁₅). We prove that if Z[α] = Z[ζ₁₅], then α + α∈Z if and only if a is equivalent to ζ₁₅. At the end of the paper, we prove that for β= ζ₁₅+ ζ₁₅² + ζ₁₅³ + ζ₁₅⁴ + ζ₁₅⁵ + ζ₁₅⁶ + ζ₁₅⁷. β is a generator of power base of Q(ζ₁₅).