Involutions in $\mathbb{Q}$.

Question

Define involution in an associative ring $k$ with identity as a map $k\rightarrow k$ mapping each $\alpha \in k$ to $\bar{\alpha}\in k$ such that

i) $\overline{\bar{\alpha}} = \alpha$

ii) $\overline{\alpha + \beta} = \bar{\alpha} + \bar{\beta}$

iii) $\overline{\alpha \beta} = \bar{\beta} \bar{\alpha}$

Using this, how can we show that the only involution in $\mathbb{Q}$ is the identity map?

Answer 1

• Show $\overline{1} = 1$
• Show $\overline{k} = k$ for $k \in \mathbb{Z}$ using ii)
• Show $\overline{p/q} = p/q$ with iii).

Answer 2

The ring $\mathbb{Q}$ is commutative, so by $iii)$ your involution is actually a ring automorphism, and they are known to be trivial on $\mathbb{Q}$.