The Möbius function and the residue theorem

A classical conjecture of Bouniakowsky says that a non-constant irreducible polynomial in Z[T] has infinitely many prime values unless there is a local obstruction. Replacing Z[T] with κ[u][T], where κ is a finite field, the obvious analogue of Bouniakowsky's conjecture is false. All known counterexamples can be explained by a new obstruction, and this obstruction can be used to fix the conjecture. The situation is more subtle in characteristic 2 than in odd characteristic. Here, we illustrate the general theory for characteristic 2 in some examples.