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I just came across the "Freshman's dream"-lemma, i.e. that if we have a field $K$ such that $char(K)=p$, then $(a+b)^p = a^p+b^p$.

But I wonder whether this generalizes for more summands, i.e. if it holds for $(a+b+c)^p = a^p + b^p+c^p$ and so on. I would not have expected it to hold but I've tried a couple of examples and it did hold for all of them, so I'm a bit surprised.

Does someone know whether this holds for more summands and if yes, how to prove it?

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    Induction is welcome here. – 2017-01-21

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Yes: $(a+b+c)^p=((a+b)+c)^p=(a+b)^p+c^p=a^p+b^p+c^p$ if $\mathrm{char}(K)=p$. This argument can be generalized to any number of summands using induction.