Power Integral Bases Of Cyclotomic Field Q(ζ₂₀)

A Galois number L is said to have a power basis if its ring of integers is of the form Z[α] for someα∈L. In this caseαis called a generator of power basis in L. Letαandβbe generators of two power bases in L,αandβis called equivalent ifβ= m±σ(α) for some m∈Z,σ∈Gal(L/Q). In this paper, we discuss the generators of power integral bases of cycloyomic field Q(ζ₂₀).Z[ζ₂₀] is the ring of integers of the cyclotomic field Q(ζ₂₀). soζ₂₀ generates a power integral basis for Q(ζ₂₀). Letαis another generator of a power integral basis of cyclotomic field Q(ζ₂₀). I prove that ifα+α(?) Z, thenαis equivalent toζ; Ifα+α∈Z, thenαis equivalent to 1/1+ζ₂₀ .Therefore, we can get all the generators of power integral bases for the cyclotomic field Q(ζ₂₀) under equivalence.