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Let $\mathbb{Z}_{p}$ denote the ring of $p$- adic numbers.

  • How can I prove that every elements of $1+8\cdot \mathbb{Z}_{2}$ is a square.

I am not comfortable in working $\mathbb{Z}_{p}$'s. So a detailed solution would be of great help and I would learn in future as to how to deal with such problems.

Thanks.

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    @PantelisDamianou, read the first line of the question. This is not about the residue class ring of integers, i.e. $\mathbb{Z}/2\mathbb{Z}$.2012-07-14

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Indeed this is a Hensel's Lemma calculation, in fact a very standard one in the theory of local fields. It often goes under the name Local Square Theorem. For a statement and proof, see e.g. Lemma 2.11 of these notes.

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    L.Clark: Thanks a lot.2012-07-14