10
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find total number of maximal ideals in $\mathbb{Q}[x]/\langle x^4-1\rangle$.

Let $J=\langle x^4-1\rangle$, $R=\mathbb{Q}[x]$. I want to use $(R/J)/(I/J)\simeq R/I$, where $I $ is ideal of $R$ which contain $J$. Then $R/I$ is field, and $R$ is a principal ideal domain. Let $I=\langle f(x) \rangle$ hence $f(x)$ must be irreducible in $R$, so only choice for $f(x)$ are $x-1,x+1,x^2+1$.

So answer should be $3$. Is it right explanation? and better method

thanks in advance

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  • 2
    $\begingroup$ That's perfect, and exactly the right way to go. You should also quote the fourth isomorphism theorem which says that those ideals are in bijection, the third isomorphism theorem (which you quote) tells you there are at least those three, but the fourth tells you there are no others. $\endgroup$ – Adam Hughes Feb 6 '17 at 3:53
  • $\begingroup$ @AdamHughes ya exactly.thanks $\endgroup$ – Eklavya Feb 6 '17 at 4:06
3
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That is all correct. (To get this out of the unanswered queue.)

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Björn Friedrich Feb 15 '17 at 5:05
  • $\begingroup$ @BjörnFriedrich The question was "is this (complete solution in the body of the post) correct?" So this IS an appropriate answer. Be sure to consult the original post in a review like this. $\endgroup$ – rschwieb Feb 15 '17 at 11:46

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