First, we need to show that $K$ is closed under addition and multiplication.
Let $x, y \in K$. Then, by definition, $x^p = x$ and $y^p = y$. We want to show that $(x+y)^p \in K$ and $(xy)^p \in K$.
Using the binomial theorem, we have
\[(x+y)^p = \sum_{k=0}^p
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