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Let $p$ be an odd prime number.

As a corollary of Hensel's lemma, we know that for $m\in \mathbb{N}$ there is a primitive $m$-th root of unity iff $m|(p-1)$.

Hence, if $p\equiv 1$ mod 4, we have a $2^n$-th root of unity in $\mathbb{Q}_p$, for some $n\geq 2$.

My question is the following: once we have established the existence of such roots, how can we calculate the exact number of 2-power roots of unity in the $p$-adic field?

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    $\begingroup$ If $p \nmid 2n$ then $X^{2n}+1$ has $2n$ different roots in $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}_p}$ and $\overline{\mathbb{F}_p}$. Those in $\mathbb{Q}_p$ are the ones corresponding to the solutions in $\mathbb{F}_p$ : there are $gcd(p-1,2n)$ of them. $\endgroup$
    – reuns
    Sep 4, 2017 at 19:47

1 Answer 1

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For $p$ an odd prime, the roots of unity in $\Bbb Q_p$ correspond to the nonzero elements of $\Bbb F_p$ and form a cyclic group of order $p-1$. So the number of $2$-power roots of unit in $\Bbb Q_p$ is the largest power of $2$ dividing $p-1$.

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  • $\begingroup$ How many roots of unity in $\Bbb Q_2(\sqrt{-3})$ ? Is there only one root of unity according to your answer? $\endgroup$
    – MAS
    Oct 28, 2019 at 12:36

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