Distinguished polynomial ($f(T)=T^d+a_{d-1}{T}^{d-1}+\cdots +a_0,a_i\in p{\mathbb{Z}}_p$)

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.