March 18th, 2018, 09:14 AM  #1 
Member Joined: Sep 2011 Posts: 98 Thanks: 1  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. 
March 18th, 2018, 02:35 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,642 Thanks: 625 
The thumbnails are very hard to read.

March 18th, 2018, 02:45 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra 
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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