Congruences on Rational Numbers
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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:
we know that both these relations are congruent 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 $\RR \ne \Delta_\Q$.
\(\ds \RR\) | \(\ne\) | \(\ds \Delta_\Q\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists r, s \in \Q: \, \) | \(\ds r\) | \(\ne\) | \(\ds s \land r \RR s\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists h \in \Q, h \ne 0: \, \) | \(\ds h\) | \(=\) | \(\ds r - s\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds h\) | \(\RR\) | \(\ds 0\) | as $\RR$ is a congruence relation for $+$ |
Then:
\(\ds \forall x \in \Q: \, \) | \(\ds \paren {x / h}\) | \(\RR\) | \(\ds \paren {x / h}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\paren {x / h} \times h}\) | \(\RR\) | \(\ds \paren {\paren {x / h} \times 0}\) | as $\RR$ is a congruence relation for $\times$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\RR\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \eqclass 0 \RR\) | \(=\) | \(\ds \Q\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \RR\) | \(=\) | \(\ds \Q \times \Q\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.3$