Dedekind's criterion clarification

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Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf)

Is it true that the prime ideals $\mathfrak{p}_i := (p, h_i(\alpha))$ are distinct? Why?