TITLE: An identity in Ring of characteristic $p$ prime
QUESTION [3 upvotes]: Is it true that in a ring of prime characteristic $p$ results that
$(x-1)^{p-1}=1+x+x^2+...+x^{p-1}$ ?
If this is not true in general, the assumption that $x$ is a nilpotent element (let's say $x^{p^n}=0$) make it works?

REPLY [3 votes]: Yes this is true. As the polynomial ring $\Bbb F_p[x]$ is a domain we may cancel $(x-1)$ from
$$\color{Green}{(x-1)}(x-1)^{p-1}=(x-1)^p=x^p-1=\color{Green}{(x-1)}(x^{p-1}+\cdots+x+1).$$
Since the identity holds for polynomials in an indeterminate, it must also hold for any value of $x$ in an $\Bbb F_p$-algebra (even if $x-1$ were not cancellable in it like it is in $\Bbb F_p[x]$).

REPLY [2 votes]: The identity does indeed hold in general.  One way to see this is from the binomial coefficient identity
$$\binom{p-1}{n}\equiv (-1)^n\pmod p$$
To see that this identity holds, notice that
$$\binom{p-1}{n}=\frac{(p-1)(p-2)\ldots(p-n)}{1\cdot2\cdot\ldots\cdot n}\equiv\frac{-1\cdot-2\cdot\ldots\cdot-n}{1\cdot2\cdot\ldots\cdot n}\pmod{p}$$

REPLY [2 votes]: Yes this is true, the binomial theorem gives
$$(x - 1)^{p-1} = \sum_{k = 0}^{p-1} \binom{p - 1}{k}(-1)^{p - 1 - k}x^k$$
and in characteristic $p$ one can show that
$$\binom{p - 1}{k} = (-1)^k$$
and
$$(-1)^{p - 1 - k} = (-1)^k$$
so the above simplifies to
$$(x - 1)^{p-1} = \sum_{k = 0}^{p-1}x^k.$$
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