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Identify the following ring: $$\mathbb{Z}[x]/(x^2 − 3, 2x + 4).$$

I am very confused on how to go about solving this problem, and mainly about ideals in general. I know that $x^2-3=0$ and $2x+4=0$ in this ring. But how do I use these ideals to be able to identify the ring. Thank you.

user26857
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Jacob Rodgers
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  • One place to start: can you deduce any interesting equations from $x^2 - 3=0$ and $2x+4=0$? – Andrew Dudzik May 04 '16 at 19:39
  • implies that x^2=3 and 2x=-4. Do I want to find some linear combination of the two and set that equal to zero?? – Jacob Rodgers May 04 '16 at 19:43
  • @JacobRodgers Whether you want to or not, you should! Linear combinations won't be enough, but can you think of other ways to combine those two equations? – Andrew Dudzik May 04 '16 at 19:46
  • @MikhailGoltvanitsa I think $(x^2-3, 2x+4)$ is the ideal generated by $x^2-3$ and $2x+4$. Interestingly, adding them yields $x^2+2x+1=(x+1)^2$. Maybe that could help? – Noble Mushtak May 04 '16 at 19:47
  • multiply 2x=-4 by 2x to get that 2(x^2)=-4x?? Which would imply that -2x=3?? if we substitute back into the first equation?? – Jacob Rodgers May 04 '16 at 19:49

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