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

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 the multiplicative quotient group $\Q_p^\times \mathop/ \paren{\Q_p^\times}^2$ has order $4$:

$\exists c \in \Q_p^\times \setminus \paren{\Q_p^\times}^2 : \set{1, p, c, cp}$ is a transversal

Corollary

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

Proof

By definition of field:

$\Q_p^\times = \Q_p \setminus \set{0}$ is an abelian group

From Group of Units is Group:

$\struct{\Q_p^\times, \times}$ is a subgroup of $\struct{\Q_p^*, \times}$

From Power of Elements is Subgroup:

$\struct{\paren{\Q_p^\times}^2, \times}$ is a subgroup of $\struct{\Q_p^\times, \times}$

By definition of quotient group, the quotient group $\Q_p^\times \mathop/ \paren{\Q_p^\times}^2$ can be formed.

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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 $37$