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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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You don't have to be sorry, but please don't use the imperative "show" either. How far do you get yourself on this problem? Where are you stuck? – Gregor Bruns Nov 19 '12 at 12:58
    
Again, please read the FAQ and follow the rules of the site. meta.math.stackexchange.com/questions/1803/… – Noah Snyder Nov 19 '12 at 22:35

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.

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