Exercise 2.2.17* (Coefficients of a power series divisible by $p$)

Question

Let the number $c_i$ be defined by the power series identity
$$(1+x+\cdots+x^{p-1})/(1-x)^{p-1} = 1 +c_1x+c_2x^2+\cdots.$$
Show that $c_i \equiv 0 \pmod p$ for all $i\geq 1$.

Richard Ganaye · Accepted Answer

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\begin{proof} 
Consider 
\begin{align*}
f(x) &= (1+x+\cdots+x^{p-1})\frac{1}{(1-x)^{p-1}}\
&= \frac{1 -x^p}{(1-x)^p}\
& = \sum_{i=0}^\infty c_ix^i  = 1 + c_1x+ c_2x^2 +\cdots
\end{align*}
($c_0 = f(0) = 1$).

We can consider these equalities in the real (or complex) field, and the series as convergent series for $|x| <1$. But it is more convenient here to consider $f(x)$ as a formal series, so that $f(x) \in \R[[x]]$, and more precisely $f(x)$ is in the subring $A = \Z[[x]]$ of formal series with integer coefficients, because $1/(1-x) = \sum_{i=0}^\infty x^i \in A$ so that $c_i \in \Z$ (we can compute $c_i$ using Exercise 1.4.7).

\bigskip
Here we denote $[c]_p}$ the class modulo $p$ of some integer $c$ (and $1 = [1]_p,\ 0 = [0]_p$).

Then we reduce modulo $p$ the equality in $\Z[[x]]$
$$1-x^p = (1-x)^p \sum_{i=0}^\infty c_ix^i .$$
This comes down to apply the ring homomorphism
$$
\varphi
\left\{
\begin{array}{ccl}
\Z{[}{[}x{]}{]} & 	o & \F_p{[}{[}x{]}{]}\
\sum_{i = 0}^\infty a_i x^i & \mapsto & \sum_{i = 0}^\infty [a_i]_p x^i .
\end{array}
ight.
$$
We obtain
$$1-x^p = (1-x)^p \sum_{i=0}^\infty [c_i]_p x^i  \in \F_p[[x]].$$
Using $\binom{p}{k} \equiv 0\ (1 \leq k \leq p-1)$, we obtain the equaliy in $\F_p[[x]]$
$$(1-x)^p = 1-x^p.$$
(This is true is $p$ is an odd prime, and also true if $p = 2$, since $1 = -1$ in $\F_2$.)
Thus 
$$1-x^p = (1-x^p) \sum_{i=0}^\infty [c_i]_px^i  \in \F_p[[x]].$$
Moreover $1 -x^p$ is invertible in $\F_p[[x]]$, since
$$(1 -x^p)\sum_{i=0}^\infty x^{pi} = 1.$$
Therefore, simplifying by $(1-x^p)$,
$$1 =  1 + [c_1]_p x + [c_2]_p x^2+ \cdots,$$
so
$$0\equiv c_1 \equiv c_2 \equiv \cdots \pmod p.$$ 
This shows that $c_i \equiv 0 \pmod p$ for all $i\geq 1$.
\end{proof}

Exercise 2.2.17* (Coefficients of a power series divisible by $p$)

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Exercise 2.2.17* (Coefficients of a power series divisible by $p$)

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