Suppose $x$ and $y$ belong to an integral domain of prime characteristic $p$. How can I prove that
$(x+y)^{p^n} = x^{p^n} + y^{p^n}$
for all positive integers $n$? Please help.
Suppose $x$ and $y$ belong to an integral domain of prime characteristic $p$. How can I prove that
$(x+y)^{p^n} = x^{p^n} + y^{p^n}$
for all positive integers $n$? Please help.
Start by demonstrating that $(x+y)^p=x^p+y^p$. You can do this by writing
$(x+y)^p=\sum_{k=0}^p \binom{p}{k}x^ky^{p-k}$
and showing that $p \mid \binom{p}{k}$ unless $k=0,p$. Then the result follows by induction.