Congruences on Rational Numbers

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


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

Proof
From:
 * Diagonal Relation is Universally Compatible and
 * Trivial Relation is Universally Congruent

we know that both these relations are compatible with both addition and multiplication on $\Q$.

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

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

Then: