April 25th, 2017, 02:01 PM  #1 
Newbie Joined: Apr 2017 From: United Kingdom Posts: 1 Thanks: 0  Homework Problem
Hello, I am a bit stuck on this question and can't start it.. Any help is very welcome, thanks in advance. Q: In the ring R=ℚ[x]/(x^5) find the inverse element for (12(x^2)+x^3). 
April 25th, 2017, 04:21 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,819 Thanks: 750 
Q[x]/x^5, if I understand your notation, correctly, can be interpreted as the field of all polynomials, with rational coefficients, of degree 5 of less with multiplication defined as regular polynomial multiplication except that we drop all powers of x larger than 4. To find the multiplicative inverse of 1+ 2x^2+ x^3, we look for ax^4+ bx^3+ cx^2+ dx+ e such that this multiplication gives 1. That is, we look at (x^3+ 2x^2+ 0x+1)(ax^4+ bx^3+ cx^2+ dx+ e)= ax^7+ (2a+ b)x^6+ (2b+ a)x^5+ (a+ 2c+ d)x^4+(c+2e)x^2+ dx+ e= (a+ 2c+ d)x^4+ (b+ 2d+ e)x^3+(c+2e)x^2+ dx+ e since we drop the x^7, x^6, and x^5 terms. We want that to be equal to 1 so we must have a+ 2c+ d= 0, b+ 2d+ e= 0, c+ 2e= 0, d= 0, and e= 1. Solve those five equations for a, b, c, d, and e. 

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