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Let $p \ne 2$ be prime number and denote by $\zeta_p$ the p-th root of unity. It's well known that $K = \mathbb{Q}_p(\zeta_p)$ has $t=1 - \zeta$ as prime element (generator of the Ideal $P_K = \{ x\in \mathbb{Q} | |x|<1 \}$ in the ring of integers $O_K= \{x\in \mathbb{Q} | |x| \leq 1\}$.

Let $\sigma:\zeta_p \to \zeta_p^g$ be an automorphism of $K/\mathbb{Q}_p$ ($1\leq g\leq p-1$).

Show $\sigma(t) \equiv gt \;(t^2)$ (it means $\sigma(t)-gt \in (t^2)$, the ideal generated by $t^2$)

and $t^{-p+1} p \equiv -1 \; (t)$.

Any hints ?

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    -1 by Wilson Theorem. Thank you !2012-10-31

0 Answers 0