Let be a field. Consider the ring of formal power series in i.e. if , then where .

Multiplication is defined by .

(a) Prove that is a unit if and only if the constant term . (ex. is the inverse of )

(b) Prove that is a Euclidean domain with respect to the norm if is the

first term of that is non-zero.

(c) In the polynomial ring , prove that is irreducible.