My Math Forum Ring Theory Problem, Help highly appreciated!!

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 February 21st, 2013, 05:13 AM #1 Newbie   Joined: Feb 2013 Posts: 2 Thanks: 0 Ring Theory Problem, Help highly appreciated!! Here is the problem: Consider the factor ring F2[x]/ (a) List the elements of this ring and write out the Cayley tables for addition and multiplication in this ring. (b)Find the units and the ideals of this factor ring. So far, I have found the elements to be {0, 1, x, x+1, x^2, x^2+1, x^2+x, x^2+x+1}, but I am struggling with the Cayley tables and the units and ideals. Does anyone know how to find the solutions to these problems? Thankyou
 February 21st, 2013, 03:48 PM #2 Member   Joined: Jan 2013 Posts: 93 Thanks: 0 Re: Ring theory problem I’ll do an example. Suppose you want to find $(x^2+x+1)+(x^2+x)$ and $(x^2+x+1)(x^2+x)$. Addition is straightfoward: $(x^2+x+1)+(x^2+x)=2x^2+2x+1\equiv1$. For multiplication, proceed as follows. 1. Multiply the terms out like ordinary polynomials (don’t worry about reducing the coefficients mod 2 at this stage): $(x^2+x+1)(x^2+x)=x^4+2x^3+2x^2+x$ 2. Divide by $x^3+x$: $x^4+2x^3+2x^2+x=(x+2)(x^3+x)+x^2-x$ 3. Finally reduce the coefficients of the remainder mod 2: $x^2-x\equiv x^2+x$ Hence $\fbox{(x^2+x+1)(x^2+x)=x^2+x}$ in $\mathbb{F}_2[x]/\langle x^3+x\rangle$.
 February 23rd, 2013, 07:42 AM #3 Newbie   Joined: Feb 2013 Posts: 2 Thanks: 0 Re: Ring Theory Problem, Help highly appreciated!! Thankyou very much! That was really helpful

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