Exercise 3, § 8

Lie Groups and Lie Algebras, II, 206

Suppose that $K$ contains a primitive $p$ -th root of unity $w$ . We write $\pi = w-1$ . Using the formula

w^i - 1 = \pi(1 + w + \cdots + w^{i-1})

show that

v(w^i - 1) = v(\pi)

for

1 \leq i \leq p-1

and that

\frac{w^i - 1}{\pi} \equiv i \pmod{\pi}.

Deduce, by means of the formula

p = \prod_{i=1}^{p-1} (w^i - 1)

, that

\pi

is admissible (Exercise 1).

Solution

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