Binomial theorem in a field of cardinality q=p^r

Question

If $a$ and $b$ are roots of the polynom $X^q - X$, how do you show that $$(a+b)^q = a^q + b^q$$

Thanks!

Answer 1

In any field of characteristic $p$, $(a+b)^q=a^q+b^q$ for any $a$ and $b$ and any $q$ which is a power of $p$. This follows by induction from $(a+b)^p=a^p+b^p$ (which follows from the binomial theorem).