= Algebraic closure of the p-adic numbers is not complete
Choose compatible primitive <roots of unity> $\zeta_{p^{n+1}}$. The partial sums $A_N=\sum_{n=0}^Np^{2n}\zeta_{p^{n+1}}$ form a <Cauchy sequence>. Each $A_N$ generates $\mathbb Q_p(\zeta_{p^{N+1}})$: for a nonidentity cyclotomic automorphism with $r=v_p(a-1)$, the first changed term has valuation $2r+1/(p-1)$, strictly below every later changed term. Thus the conjugate distance is at least $p^{-2N-1/(p-1)}$. The tail bound is $p^{-2N-2}$, strictly smaller. If the limit were algebraic, <Krasner's lemma> would place every one of these unbounded-degree cyclotomic fields in one finite extension, a contradiction.
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