p-th powers in $\mathbb{Q}_p(\zeta)$

Question

Let $p\ne 2$. Consider $\mathbb{Q}_p(\zeta)$ where $\zeta^p=1$.

1. Show that $\mathbb{Q}_p(\zeta)/\mathbb{Q}_p$ is completely ramified and that $\lambda = 1-\zeta$ is a prime element.
2. Show $\lambda^{1-p}p=-1(\lambda)$
3. Show that if $a\in\mathbb{Q}_p(\zeta)$ is a unit with $a=1(\lambda)$ then there exists $m\in\mathbb{Z}: \zeta^ma=1(\lambda^2)$
4. Show that a unit $b\in\mathbb{Q}_p(\zeta)$, $b=1(\lambda)$ is a p-th power iff $b=1(\lambda^{p+1})$
5. If $b=1(\lambda^p)$ but $b\not=1(\lambda^{p+1})$ and $\theta^p=b$ show that $\mathbb{Q}_p(\theta,\zeta)/\mathbb{Q}_p(\zeta)$ is an unramified extension of degree $p$.

I can do 1-3 but not 4 or 5. For 4 I am trying to use Hensel's Lemma but the best I can get is when $b=1(\lambda^{2p})$ which also occurs from 3. The book gives a hint to use $f(Y)=(a+\lambda Y)^p-b$ and 3 for the other direction but I fail to see how. For 5 I am not sure how to show that this extension is unramified as the polynomial $X^p-b(\lambda)=(X-1)^p(\lambda)$ so it is not irreducible and I cannot use the usual criterion.