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Find a simpler description of $\mathbb Q[x]/(x^3 + x)$.

Since $x^3 = -x$ in the quotient space, I know all the polynomials can be reduced to the form $a + bx + cx^2$ where $a,b,c\in\mathbb Q$.

I also know that since $x^3 + x = x(x^2 + 1)$ is reducible, the ideal is not prime or maximal, so the quotient space is not a domain.

Is this as simple as it gets, or is there some other simpler description of this ring?

user26857
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fdzsfhaS
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1 Answers1

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You're on the right track, but a little more can be said: using the Chinese remainder theorem, $$ \mathbb{Q}[x]/(x^3+x)\simeq \mathbb{Q}[x]/(x)\times\mathbb{Q}[x]/(x^2+1)\simeq\mathbb{Q}\times\mathbb{Q}[i]$$ where $i^2=-1$.

carmichael561
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