Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 136 / 4 / b

Solution

Put

π_{n} = ζ_{p^{n}} - 1

Φ_{p^{n}} (1 + X) = \frac{( 1 + X ) ^{p^{n}} - 1}{( 1 + X ) ^{p^{n - 1}} - 1}

is Eisenstein at

p

. Therefore it is irreducible,

π_{n}

[Q_{p} (ζ_{p^{n}}) : Q_{p}] = φ (p^{n}) = p^{n - 1} (p - 1),

so the extension is totally ramified. It is the splitting field of

Φ_{p^{n}}

, and every automorphism is uniquely

ζ_{p^{n}} ⟼ ζ_{p^{n}}^{, a}, a \in (Z / p^{n} Z)^{\times} .

Gal (Q_{p} (ζ_{p^{n}}) / Q_{p}) ≅ (Z / p^{n} Z)^{\times} .

Restriction in the cyclotomic tower corresponds to reduction of

a

, so taking the inverse limit gives

Gal (Q_{p} (ζ_{p^{\infty}}) / Q_{p}) ≅ lim_{n} (Z / p^{n} Z)^{\times} = Z_{p}^{\times} .

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