Let $K$ be a number field and $K_{\infty}/K$ the cyclotomic $\mathbb Z_p$-extension of $K.$
My question is : How to prove that for any prime $\ell$ of $\mathbb Q$ distinct to $p$ does not decompose completely in $K_{\infty}/K$ ?
Thank you very much for any help :-).
It suffices to consider $K = \mathbf{Q}$, since for a general $K$ we have $K_\infty = K \mathbf{Q}_\infty$.
If $\mathbf{Q}_n$ is the $n$th layer of $\mathbf{Q}_\infty$ (i.e. the degree $p^n$ subextension), then the Galois group of $\mathbf{Q}_n / \mathbf{Q}$ is the quotient of $\mathbf{Z} / p^{n+1}$ by the $(p-1)$-st roots of unity; and the Frobenius at $\ell$ corresponds to $\ell \bmod p^{n+1}$. So $\ell$ splits completely in $\mathbf{Q}_n$ if and only if $\ell^{p-1} = 1 \bmod p^{n + 1}$. It's clear that for each $\ell$, this happens for only finitely many $n$.