Yes. This continues, p. of Proposals 16 and 17 in Serre: Local fields.

In fact, let's leave $ n = p ^ mn & # 39; $, where $ (n & # 39 ;, p) = 1 $. Applying Proposition 17, we see that $ mathbb {Q} _p ( zeta_ {p ^ m}) $ has an integer ring $ mathbb {Z} _p ( zeta_ {p ^ m}) $. Then, applying Proposition 16, we see that $ mathbb {Q} _p ( zeta_ {p ^ m}, zeta_ {n & # 39;}) $ has an integer ring $ mathbb {Z} _p ( zeta_ {p ^ m}, zeta_ {n & # 39;}) $. However, it is simple that $ mathbb {Q} _p ( zeta_ {p ^ m}, zeta_ {n & # 39;}) = mathbb {Q} _p ( zeta_ {n}) $ Y $ mathbb {Z} _p ( zeta_ {p ^ m}, zeta_ {n & # 39;}) = mathbb {Z} _p ( zeta_ {n}) $, so we are done.