Is ring of integers of $\mathbb{Q}_p(α) $ is $\mathbb{Z}_p[α] $ ?
If $α$ is primitive root of unity, dependentable reference reads that the titled statement holds. But in general, does the statement hold?
My try: In local fields, rings of integers is the same thing as valuation ring(I don't know hot to prove this). So, there are denominator has no prime factor $p$, so the integer ring is $\mathbb{Z}_p[α] $.