This requires knowledge of Dedekind cuts.
Let us define the cut $\beta$
$\beta = \mathbb{Q}_- \cup \lbrace 0\rbrace \cup \lbrace t: 0 with the property $ \alpha \beta = 1^*$ where $1^*$ is the multiplicative identity, i.e. $1^* = \lbrace p: p<1 \rbrace$ I don't know how to prove $\beta$ is a cut (let alone that it actually behaves like a multiplicative inverse), and I can't find anything online to help me figure it out. So, for $\beta$, how do I prove $\beta \neq \mathbb{Q}, \emptyset$ (i.e. $\beta$ consists of neither all the rationals nor the empty set). $\beta$ is closed downward: I.e. for any rational $r\in\beta$, there exists $s\in\beta$ such that $s $\beta$ has no largest element: I.e. for any rational $r\in\beta$, there exists $t\in\beta$ such that $r I literally can't even prove that $\beta$ is non-empty. Now, maybe if I get some help here I'll be able to prove the rest, but I doubt that. Thanks.