Finding root using Hensel's Lemma

Question

Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of $\mathbb{Z}_p[X]$ important?

Answer 1

For an application, coding theory is a good example. There is a theory for codes (=modules) over rings of the form $\mathbb Z/(p^s)$ which involves lifting polynomials from $\mathbb Z/(p)[x]$ to $\mathbb Z/(p^s)[x]$. On the other hand, the real numbers don't play a big role in coding theory, since you can't represent them exactly in a computer.

By the way, Hensel's Lemma not only lifts roots, but more generally also factorizations of separable polynomials. Roots correspond to factors of degree $1$, but the factors to be lifted can be of any degree.

Answer 2

The general, stronger, version of Hensel that @azimut mentions in his second paragraph can be used, for instance, to show that the valuation on $\mathbb Q_p$ extends uniquely to any finite extension of this field, and hence to any algebraic extension of $\mathbb Q_p$. This is an important fact.