Quotient Group of Quadratic Residues Modulo 2 of 2-adic Units

Theorem

Let $\Q_2$ be the $2$-adic numbers.

Let $\Q_2^\times$ denote the set of invertible elements of $\Q_2$.

Let $\paren{\Q_2^\times}^2 = \set{a^2 : a \in \Q_2^\times}$

Then the multiplicative quotient group $\Q_2^\times \mathop/ \paren{\Q_2^\times}^2$ has order $8$ with:

$\set{1, -1, 5, -5, 2, -2, 10, -10}$ is a transversal

Corollary

$\Q_2^\times \mathop/ \paren{\Q_2^\times}^2$ is isomorphic to $\Z \mathop/ 2\Z \oplus \Z \mathop/ 2\Z \oplus \Z \mathop/ 2\Z$

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 $44$