Classes of polynomials having only one non-cyclotomic irreducible factor

Classes of Polynomials Having Only One Non{cyclotomic Irreducible Factor

He noted then that the conjecture is true if n = p 1 2 or if n = p where p is a prime and r a positive integer. Calculations showed the conjecture also held for n 100. Recently, in a study of more general polynomials, the rst author [2] obtained further irreducibility results for f(x); in particular, he established irreducibility in the case that n+ 1 is a squarefree number 3 and in the case th...

Classes of Polynomials Having Only

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Irreducible Values of Polynomials: a Non-analogy

For a polynomial f(T ) ∈ Z[T ], the frequency with which the values f(n) are prime has been considered since at least the 18-th century. Euler observed, in a letter to Goldbach in 1752, that the sequence n + 1 has many prime values for 1 ≤ n ≤ 1500. Legendre assumed an arithmetic progression an+ b with (a, b) = 1 contains infinitely many primes in his work on the quadratic reciprocity law. Ther...

Cyclotomic Polynomials