| In this paper, I first give two methods with which we can determine all the power integral bases in maximal real subfields of cyclotomic fields. Then I determine all the power integral bases for (?)(ζm+ζm-1)(m = 5, 7, 8, 9, 12, 16, 20, 24) These results may be helpful to the research of the power integral bases in maximal real subfields of any cyclotomic fields.Meanwhile I prove that the equation x4-y4 = z2 has only trivial solutions in (?)[(-2)1/2],(?)[21/2]]. So it indicates that square of any nonzero integer z ∈ (?)[(-2)1/2] (respectively (?)[21/2]) can't be expressed as the difference of two quar-tics of nonzero integers in (?)[(-2)1/2)(respectively (?)[21/2]).At last I show the number n of representations of the nonzero integer z G (?)[(-2)1/2] as the difference of two squares of integers in (?)[(-2)1/2]. At the same time I show the number n of representations of the nonzero integer z ∈ (?)[(-2)1/2] as z = x2 + 2y2, where x, y∈(?)[(-2)1/2]... |
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