Composed products and module polynomials over finite fields
Abstract A general notion of composition of polynomials over a finite field was introduced by Brawley and Carlitz (1987). In their paper they developed a unique decomposition theory in two special cases of the composed product, but they left open the question of uniqueness in the general case. This paper settles this open question by proving that irreducibles do indeed decompose uniquely with respect to the general notion and, in doing so, unifies the two separate treatments of Brawley and Carlitz into a single, more general theory. The paper also offers a generalization which unifies into a single theory the notions of ‘belonging to an exponent’ and ‘belonging to a q -polynomial’, which are basic in the theory of finite fields.