Weierstrass preparation theorem

Over a complete local ring, a power series whose first unit coefficient has index $d$ is a unit times a unique monic distinguished polynomial of degree $d$. Consequently its sufficiently small zeros, counted with multiplicity, are the zeros of that polynomial.

A distinguished polynomial over ${\mathbb{Z}}_p$ is monic and has every non-leading coefficient divisible by $p$. The Weierstrass preparation theorem expresses any nonzero element of ${\mathbb{Z}}_p[[T]]$ as a power of $p$ times a unit times a distinguished polynomial.