I've been stumbling through Washington's "Introduction to Cyclotomic Fields (2nd edition)" and chapter §13.3 on Iwasawa's theorem has some claims that I am having a hard time with.
The chapter starts off with a $\mathbb{Z}_p$-extension $K_\infty/K$, so $Gal(K_n/K) \simeq \mathbb{Z}/p^n\mathbb{Z}$. For each extension $K_n$ let $L_n$ be its maximal unramified abelian p-extension.
Next comes the assumption that is used for the first half of the chapter.
Assumption: All primes which are ramified in $K_\infty/K$ are totally ramified.
The author claims that from the assumption it follows that $K_{n+1}\cap L_n = K_n$, and from this $Gal(L_n/K_n)\simeq Gal(L_nK_{n+1}/K_{n+1})$. Why is this true?
A few remarks:
There must be some prime $\mathfrak{p}$ of $K$ above $p$ that is ramified in $K_\infty/K$. This is simply because $K_\infty/K$ is an infinite abelian extension and class field theory tells us that the maximal unramified abelian extension is finite over $K$.
If $\mathfrak{p}$ is a prime that totally ramifies in $K_\infty/K$, then it also (totally) ramifies in any subextension $M/L$ with $K \subset L \subset M \subset K_\infty$. Thus $(K_{n+1} \cap L_n)/K_n$ is totally ramified at $\mathfrak{p}$, but $L_n/K_n$ is unramified everywhere, hence $(K_{n+1} \cap L_n)/K$ must also be unramified everywhere. This is only possible if $K_{n+1} \cap L_n$ is the trivial extension of $K_n$.
For any finite Galois extension $L/K$ and finite extension $M/K$, Galois theory gives a canonical identification $\mathrm{Gal}(ML/M) \cong \mathrm{Gal}(L/ L \cap M)$. Thus $\mathrm{Gal}(L_{n} K_{n+1} / K_{n+1}) \cong \mathrm{Gal}(L_n/ (L_n \cap K_{n+1})) = \mathrm{Gal}(L_n/K_n).$