A note on the Brawley-Carlitz theorem on irreducibility of composed   products of polynomials over finite fields
  We give a new proof of the Brawley-Carlitz theorem on irreducibility of the composed products of irreducible polynomials. Our proof shows that associativity of the binary operation for the composed product is not necessary. We then investigate binary operations defined by polynomial functions, and give a sufficient condition in terms of degrees for the requirement in the Brawley-Carlitz theorem. 
Akihiro Munemasa and Hiroko Nakamura
math.NT