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 $(x+a)^p=x^p+a^p$?

  • $\begingroup$ 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

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 $r=0,p$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.