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March 18th, 2018, 08:14 AM   #1
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Irreducibility

Hi all,

I have done the question in two methods. The first method is done by rational root test and the second method is by modulo p (theorem is as attached). It seems that my answers for both methods do not tally.

1. Where have I done wrong in the attached for the methods? Which is the correct presentation of answer for this question (i.e. rational test method or modulo p ?
2. How do I tell when to use rational root test method or modulo p method? When modulo p method not applicable?

Your advise is greatly appreciated. Thanks.
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March 18th, 2018, 01:35 PM   #2
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The thumbnails are very hard to read.
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March 18th, 2018, 01:45 PM   #3
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The first thing I notice is that your modulo $p$ theorem says that if for some prime $p$, $\bar f(X)$ is irreducible in $\mathbb Z_p[x]$ then $f(x)$ is irreducible in $\mathbb Q[x]$. It does not say that if $\bar f(X)$ is reducible, then $f(X)$ is reducible.
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