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Prove that a given quotient ring is not an integral domain


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.17

Solution:

(1) Evidently, p(x)=x211x+3, q(x)=8x213x+5, p(x)+q(x)=7x224x+8, and p(x)q(x)=146x2236x+90.

(2) Evidently, x32x+1=(x1)(x2+x1). Now neither x1 nor x2+x1 is zero in E, but their product is. Thus (for instance) x1 is a zero divisor in E. We found this factorization by noting that f(1)=0, so that x1 divides f(x).

(3) Evidently, xx2+2=1, so that x is a unit. To find this inverse, recall that if x is a unit, its inverse is represented by a polynomial of degree at most 2- say ax2+bx+c. Suppose xax2+bx+c=bx2+(c+2a)xa=1; comparing coefficients, we find that b=0, a=1, and c=2.


Linearity

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