Abstract
Let Fq be the finite field of q elements ( q, some power of a fixed prime number ) ; Fq[t] be the polynomial ring with one variable over Fq and Fq(t) the quotient field of Fq[t] . For any element in Fq[t], we will study Artin's conjecture for the Carlitz module in the case q=2, and for a Drinfeld Fq[t]-modules of rank 2 over Fq(t) . Each such Drinfeld module gives a new Fq[t]-modules structure on Fq(t) . It also induces a Fq[t]-modules structure on Fq[t]/(p) for almost all irreducible polynomial p in Fq[t] . We will study the Dirichlet density of all the monic irreducible polynomials in Fq[t] which make the induced Fq[t]-module on Fq[t]/(p) cyclic .