March 22nd, 2018, 09:58 AM  #11 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,453 Thanks: 106 
SDk I think I see what's bothering you. The operations which qualify F[x] as a ring treat x as a symbol, not a variable, so technically f(x+c) is undefined. About the best you can say is, that if c is in F, then evaluating f(x+c) is a legitimate member of F[x]. As for the algebra of F[x] when x is a variable, I don't know. Is it still a Ring? Another topic. But it doesn't matter.You can interpret f(x+c) as a prescription for obtaining a legitimate polynomial with x as a symbol. That works. 
March 22nd, 2018, 02:04 PM  #12  
Senior Member Joined: Sep 2016 From: USA Posts: 444 Thanks: 254 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
It appears this subtlety was (and still is) lost on some who have mistaken this exercise for a trivial calculus 1 problem about factoring polynomials.  
March 22nd, 2018, 02:44 PM  #13  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 226 Thanks: 76 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
 
March 26th, 2018, 06:42 AM  #14 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,453 Thanks: 106 
If f(x)is reducible over F, so is f(x+c) f(x)=(xa)(xb)(xd) say. Then f(x+c)=(xa+c)(xb+c)(xd+c). ie, f(x+c) irr cannot imply f(x) red because that would imply f(x+c) red. 
March 30th, 2018, 08:34 AM  #15 
Member Joined: Sep 2011 Posts: 97 Thanks: 1 
Thanks everyone for the help. I really appreciate it. I will share the outcome once the prof has reviewed his solution.


Tags 
polynomial 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Polynomial divisible by Polynomial  Nienna426  Number Theory  4  April 18th, 2015 08:14 PM 
Rewrite polynomial*2x as a sum of polynomial  king.oslo  Algebra  6  August 27th, 2013 06:11 AM 
polynomial  alexmath  Algebra  5  November 3rd, 2012 06:40 PM 
Polynomial help  Tutu  Algebra  3  June 26th, 2012 01:10 PM 
Divide polynomial without using polynomial identity  condemath  Algebra  3  September 20th, 2011 08:34 PM 