Uniformizer criterion for lower ramification groups

If

L / K

is a finite totally ramified Galois extension and

π_{L}

is a uniformizer, then

G_{i} = {σ : v_{L} (σ (π_{L}) - π_{L}) \geq i + 1}

for every

i \geq 0

.

Table of contents
- Ramification groups of the splitting field of Xp minus p over the p-adic numbers Uniformizer criterion for lower ramification groups

Ramification groups of the splitting field of Xp minus p over the p-adic numbers ( $Q_{p} (ζ_{p}, p p) / Q_{p}$ )

For odd

p

, the splitting field

K = Q_{p} (ζ_{p}, p p)

is totally ramified of degree

p (p - 1)

. With

ϖ = \frac{ζ _{p} - 1}{p p},

the lower ramification groups are

G_{0} = Gal (K / Q_{p}), G_{i} ≅ Z / p Z (1 \leq i \leq p), G_{i} = 1 (i \geq p + 1) .

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