= Ramification in a Zp-extension
At least one finite prime ramifies in a <Zp-extension>; otherwise the entire tower would lie in a finite <Hilbert class field>. Ramification can occur only over $p$: <local class field theory> makes inertia away from $p$ an image of a unit group with finite maximal pro-$p$ quotient, whereas $\mathbb Z_p$ has no nontrivial finite subgroup. Nonzero inertia is open in $\mathbb Z_p$, so it becomes total after passing to a sufficiently high finite layer.
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