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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.

  • 3
    Hint: You only really need to prove it for $n=1$. The rest follows easily by induction.2012-12-25

1 Answers 1

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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.