App1．Math．J．Chinese Univ．

A note about the cyclotomic polynomial pg ()and some related results ZHU Feng—xiang QI Wen-feng Abstract．The expression of cyclotomic polynomiM pq(茁)is concerned for a long time．A simple and explicit expression of pq ()in Z[x]has been showed．The form of the factors of pq ()over F2 and the upper，lower bounds of their Hamming weight are provided． The mth cyclotomic polynomial §1 西m ()in Izj m x)is defined to be m x) Dedekind’S inversion formula， ()=II ( d一1) ‘ ． dl n ( —e2Trik ／)．By ( ，m)=1 (1) We wil discuss the cyclotomic polynomial m ( ) in Z[x]and over F2 Ix]，where Z and F2 denote integral ring and finite field respectively． It is wel known that西m ()is an irreducible polynomial in z]with degree (仃L)，where denotes Euler function．If m0 denotes the largest square—free!factor of m，then m(X)= m 0(xm ／ 0)．Therefore，the computation of m ()can be reduced t‘O the case when m is the product of distinct primes．Obviously ， p x)=Xp 一 +X 一 + ⋯ +X+1，hence the next‘ interesting case is m = Pq．Here are t‘WO explicit examples