Multiplication of $p$-adic integers

Question

As I am reading Serre's A course in arithmetic, he defined $x\in \mathbb{Z}_p$ as an infinite sequence $x=(\ldots,x_n,\ldots, x_1)$ with $x_n\in \mathbb{Z}/p^n\mathbb{Z}$ and $x_{n+1}\equiv x_n$ mod $p^n$ for all $n$. He then says "multiplication in $\mathbb{Z}_p$ is defined coordinate by coordinate". Does this mean $$(\ldots, x_n,\ldots, x_2,x_1)\cdot (\ldots, x_n',\ldots, x_2',x_1')=(\cdots, x_nx_n',\cdots, x_2x_2',x_1x_1')?$$ Or maybe can someone give a clearer explanation of how multiplication works in $\mathbb{Z}_p$?

Answer 1

Yes indeed, you have understood correctly that this means $$(\ldots, x_n,\ldots, x_2,x_1)\cdot (\ldots, x_n',\ldots, x_2',x_1')=(\cdots, x_nx_n',\cdots, x_2x_2',x_1x_1'),$$ where multiplication in eachthe $n$-th coordinate is multiplication in $\Bbb{Z}/p^n\Bbb{Z}$, so if the $n$-th coordinates are $x_n,x_n'\in\Bbb{Z}/p^n\Bbb{Z}$ then the $n$-th coordinate of the product is $x_nx_n'\in\Bbb{Z}/p^n\Bbb{Z}$. Note that the product again satisfies $x_{n+1}x_{n+1}'\equiv x_nx_n'\mod{p^n}$ for all $n$.

This construction of the $p$-adic integers is an example of an inverse limit, which is a very general way of creating a new (larger) object from a partially ordered collection of (smaller) objects with morphisms between them.

Another way to view $p$-adic integers and their arithmetic, is to view a $p$-adic integer $x=(\ldots,x_n,\ldots,x_2,x_1)\in\Bbb{Z}_p$ as an 'infinitely long' integer in base $p$ $$x=\ldots+a_np^n+\ldots a_1p+a_0=\sum_{k\geq0}a_kp^k,$$ with $a_n\in\{0,\ldots,p-1\}$ such that for every $n$ you have $$\sum_{k=0}^{n-1}a_kp^k\equiv x_n\pmod{p^n}.$$ Then multiplication of two $p$-adic integers $x$ and $x'$ is the same as for regular integers, except now infinitely long: \begin{eqnarray*} xx'&=&\big(\ldots+a_np^n+\ldots a_1p+a_0\big) \big(\ldots+a_n'p^n+\ldots a_1'p+a_0'\big)\\ &=&+\ldots\big(\sum a_ka_{n-k}'\big)p^n+(a_0a_1'+a_1a_0')p+a_0a_0'. \end{eqnarray*} This can again be expressed in the form $xx'=\sum_{k\geq0}b_kp^k$ with $b_k\in\{0,\ldots,p-1\}$ after writing each coeficient $\sum a_ka_{n-k}'$ in base $p$, and carrying over where necessary.

Another way to construct the $p$-adic integers is as the ring of integers of $\Bbb{Q}_p$, the $p$-adic numbers, which is the completion of $\Bbb{Q}$ with respect to the $p$-adic absolute value defined by $|x|_p=p^{-n}$ if $x=p^n\tfrac ab$ with $a$ and $b$ coprime to $p$. But the arithmetical properties of $\Bbb{Z}_p$ are not as immediate from this perspective.