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I'm sory, I posted another problem. Show that $\mathbb{Z}_{11} [x]/\langle x^2-2\rangle $ and $\mathbb{Z}_{11} [x]/\langle x^2-3\rangle$ are not isomorphic

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    Again, please read the FAQ and follow the rules of the site. http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question2012-11-19

1 Answers 1

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The following steps lead to a solution:

  1. $x^2 - 2$ has no solutions over the field of $11$ elements, while $x^2 - 3$ has a solution (namely $x = 5$).

  2. The former ring is therefore a field, while the latter ring is not a field.

  3. Conclude.