Inverse of Multiplicative Inverse

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Theorem

Let $\left({F, +, \times}\right)$ be a field whose zero is $0_F$.

Let $a \in F$ such that $a \ne 0_F$.

Let $a^{-1}$ be the multiplicative inverse of $a$.


Then $\left({a^{-1}}\right)^{-1} = a$.


Proof

The multiplicative inverse is, by definition of a field, the inverse element of $a$ in the multiplicative group $\left({F^*, \times}\right)$.

The result then follows from Inverse of Group Inverse.

$\blacksquare$


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