Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 28 / 4 / a

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- Solution a

Solution

Expand the translated cyclotomic polynomial:

\frac{( 1 + T ) ^{p} - 1}{T} = p + (2 p) T + \dots + (p - 1 p) T^{p - 2} + T^{p - 1} .

Every nonleading coefficient is divisible by

p

, and the constant term is not divisible by

p^{2}

. The polynomial is Eisenstein, hence irreducible. Its root

λ = ζ_{p} - 1

is a uniformizer of

E = Q_{p} (ζ_{p})

, and the Eisenstein polynomial valuation criterion gives

[E : Q_{p}] = e (E / Q_{p}) = p - 1.

Thus this local cyclotomic extension is totally ramified.

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