User Name Remember Me? Password

 Abstract Algebra Abstract Algebra Math Forum

 December 5th, 2007, 06:59 PM #1 Newbie   Joined: Oct 2007 Posts: 3 Thanks: 0 ring Let R be any ring, let s be an element in R, and suppose we have some f(X)∈R[X] with f(s)=0, and D[f(X)] also giving value 0 at s. Prove that f(X)=g(X)(X-s)^2 for some g(X)∈R[X]. (hint. divide f(X) by (X-s)^2 with remainder, and compute D[f(X)] from that expression.) i understand the hint, but i am not sure what to do from there. December 6th, 2007, 01:21 PM   #2
Member

Joined: Nov 2007

Posts: 50
Thanks: 0

Re: ring

Quote:
 Originally Posted by Frazier001 Let R be any ring, let s be an element in R, and suppose we have some f(X)∈R[X] with f(s)=0, and D[f(X)] also giving value 0 at s. Prove that f(X)=g(X)(X-s)^2 for some g(X)∈R[X]. (hint. divide f(X) by (X-s)^2 with remainder, and compute D[f(X)] from that expression.) i understand the hint, but i am not sure what to do from there.
You need to know the rules how to manipulate with the formal derivative.

By long division you get the identity
f(x)=g(x)(x-s)^2+r(x).
What is the degree of r(x)?
Can you express from this equality (and the known rules for the formal derivative) Df(x)?
Using the fact that Df(x)=0, what you can say about r(x)? Tags ring Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post sebaflores Abstract Algebra 1 October 27th, 2013 04:29 PM Mathew Abstract Algebra 5 August 29th, 2010 08:53 PM tinynerdi Abstract Algebra 4 April 4th, 2010 10:17 PM cgouttebroze Abstract Algebra 5 August 14th, 2008 12:04 PM stf123 Abstract Algebra 3 December 7th, 2007 07:47 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top       