p-adic numbers are Taylor expansions

Look at $P(x) = x-p$.  Fixing $p$ fixes the P(x).

$P(x)$ is a prime in the ring $\mathbb{C}[x]$.  We can build any polynomial out of these irreducible components.

Fix a prime, and expand your function:

$$f(x) = f(p) + f'(p)(x-p) + \frac{f”(p)}{2!}(x-p)^2 + …$$

If first few functions in the taylor series of two functions agree, than the functions are close, i.e., two functions are close if their difference is divisible by high powers of the monomial.

Let’s look at the integers, replacing primes of polynomials with primes of integers, fix a prime, and expand your function:

$$n = n(p) + n'(p)p + \frac{n”(p)}{2!}p^2 + …$$

Where $n(p), n'(p), \frac{n”(p)}{2!}, …$ are formal symbols, corresponding to the last coefficient, second to last coefficient, third to last, and so on and so forth.

Two integers are close if their expansions are divisible by high powers of p. The ring of all such integer expansions has only one maximal ideal, $p$.

This is how Hensel originally came upon the definition of the p-adics!