The cyclotomic polynomials are used in many parts of math-ematics. Several works have already treated them. Lenstra in[7], used the cyclotomic polynomials in discrete logarithm cryp-tosystems over Finite Fields. In 1883, Migotti [8] showed thatall coe cients in p:q the p:q-cyclotomic polynomials are inf??1; 0; 1g (where p and q are distincts primes numbers). Later,Lam [6], gave a quick and naturel construction of p:q. Forn 2 be an integer, let A(n) denote the maximum of the ab-solute value of the coe cients of n. Beiter [2], characterizedthe pairs p and q in n = 3:p:q such that no coe cient of abso-lute value 2 can occur in n. Beiter [3] conjectured that, forall p 1 then for any prime p, A(p:n) 1and gave an in nite family of cyclotomic polynomials of degrefour.In this paper, we aim to characterize some in nite familiesof ternary cyclotomic polynomials, to characterized an in -nite family of cyclotomic polynomials of degre four and using some conjecture and theorems to give some others important results.
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