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

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:

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

Proof

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Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.7$ Hensel's Lemma and Congruences: Exercise $44$