Let $R = \{a+b \sqrt 2| a,b $ integers$\}$.
Let $M = \{a+b \sqrt 2| a,b $ integers and $5|a $ and $ 5|b\}$ be its ideal.
How would you write out $R/M$ in this form?
2026-04-02 16:02:41.1775145761
What do elements of this quotient ring look like and why?
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$R/M$ is the set $[a+b\sqrt 2]$ of equivalence classes. Since the addition in the quotient ring extends addition in $R$, this can be written as $[a]+[b]\sqrt 2$, so $R/M$ is in effect just $a+b\sqrt 2: a,b\in \mathbb{Z}/5\mathbb{Z}$.