Residue characteristic $p$ at the primes above $p$: $\|p\| < 1$ in $K_{\mathfrak{p}}$, and the $\mathbb{Z}_p$-module structure on its principal units

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Residue characteristic $p$ at the primes above $p$: $\|p\| < 1$ in $K_{\mathfrak{p}}$, and the $\mathbb{Z}_p$-module structure on its principal units · Prove2Me