Power Integral Bases For Cyclotomic Field Q(ζ₂₄)

In this paper, we discuss the generators of power integral bases of cycloyomiefield Q(ζ₂₄)-In Chapter 1, I introduce some basic knowledge , lemmas, and theorems, which are needed in chapter 2 and chapter 3. At last, I show the power integral bases in the maximal real subfield Q(ζ₂₄ +ζ₂₄^-1) of Q(ζ₂₄), and power integral bases to relative quartic extention field Q(ζ₂₄)In chapter 2,I prove that: For any algebraic integerα,∈Z[ζ₂₄],α+α|- (?) Z, Z[α] = Z[ζ₂₄] if and only ifαis equivalent toζ₂₄.In chapter 3, I prove that ifα+α|-∈Z, Z[α] = Z[ζ₂₄] if and only ifαis equivalent toη=1/(1-ζ₂₄) . In a word, we can get all the generators of power integral bases under equivalently.