Explicit Factorization Of A Class Of Polynomials Over Finite Field And Their Applications

LetqF be a finite field of odd order q. In this thesis, we get the explicit factorization of some polynomials by using the polynomial decomposition theorems, and then we use these results to determine the dimension and minimum Hamming weight of a class of irreducible cyclic codes.Firstly, we study the factorization of special polynomial xⁿ-1 over finite field, where a,b,c are positive integers and p,r are odd prime divisors of q-1. It is shown that all the irreducible factors of x^?2a?pbrc-1 over IF_q are either binomials or trinomials. In general, denote by v_p?m? the degree of prime p in the standard decomposition of the positive integer m. Suppose that every prime factor of m divides q-1, one has?1? if ??? holds true for every prime number p| q-1, then every irreducible factor of x^m- in IF_q is a binomial;?2? if q???3?mod 4?, then every irreducible factor of x^m-1 is either a binomial or a trinomial.Secondly, we first study the primitive idempotents over ???, where l is an integer that makes all the irreducible factors of xⁱ- are binomials.Lastly, we can get a class of minimal cyclic codes by the primitive idempotents. Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length l over F^q.