Absolute Value of Multiplicative Identity is 1

Theorem

• For any Norm Function $|\cdot |:\mathcal{D}\to \mathbb{R}$ of given Integral Domain/Field
• With multiplicative identity $\overrightarrow{1}$
• $|\overrightarrow{1}|=1\in \mathbb{R}$

Proof

• Let $a$ be the multiplicative inverse of 1
• Then, $1\ast a=1$
• $⟹1\ast |a|=|1|$
• $⟹|1|\ast |a|=|1|$
• $⟹|1|\ast |a|-|1|=0$
• $⟹|1|(|a|-1)=0$
• $⟹|a|=1$
• As $a$ is the multiplicative inverse of $1$, $|1|=1$, as $|1|\ne 0$ since $1\ne 0$