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I know that if $F$ is a field of characteristic $p$, then for all $a,b \in F$, $(a+b)^p=a^p+b^p$.

But what if now we have a polynomial say $x+a \in F[x]$. Is $(x+a)^p=x^p+a^p$?

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  • $\begingroup$ Yes, all the binomial coefficients $\binom{p}{k}$, for $k=1,2,...,p-1$, are multiples of $p$ in that case. $\endgroup$
    – user752802
    Mar 1 '20 at 19:39
  • $\begingroup$ If $R$ is a commutative unital ring where $p=0$ then $$(a+b)^p= a^p+b^p$$ You can replace unital by $\forall a\in R,pa=0$ it works too. When $R$ is non-commutative it fails, eg. in $M_2(\Bbb{F}_p)$. $\endgroup$
    – reuns
    Mar 1 '20 at 19:46
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Yes. Note that $(x+a)^p=\sum_{r=0}^p\binom{p}{r}x^ra^{p-r}=x^p+a^p$ because $p$ divides $\binom{p}{r}$ except when $r=0,p$.

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