Quotient Group of Quadratic Residues Modulo p of P-adic Units/Corollary

Theorem

Let $\Q_p$ be the $p$-adic numbers for some prime $p \ne 2$.

Let $\Q_p^\times$ denote the set of invertible elements of $\Q_p$.

Let $\paren{\Q_p^\times}^2 = \set{a^2 : a \in \Q_p^\times}$

Then:

$\Q_p^\times \mathop/ \paren{\Q_p^\times}^2$ is isomorphic to $\Z \mathop/ 2\Z \oplus \Z \mathop/ 2\Z$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.7$ Hensel's Lemma and Congruences: Exercise $37$