Congruences on Rational Numbers

Theorem
There are only two congruence relations on the field of rational numbers $\left({\Q, +, \times}\right)$:


 * $(1): \quad$ The diagonal relation $\Delta_\Q$
 * $(2): \quad$ The trivial relation $\Q \times \Q$.

Proof
From: we know that both these relations are compatible with both addition and multiplication on $\Q$.
 * Diagonal Relation is Universally Compatible and
 * Trivial Relation is Universally Congruent

Now we need to show that these are the only such relations.

Let $\mathcal R$ be a congruence on $\Q$, such that $\mathcal R \ne \Delta_\Q$.

Then: