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February 21st, 2013, 05:13 AM   #1
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Ring Theory Problem, Help highly appreciated!!

Here is the problem:

Consider the factor ring F2[x]/<x^3 +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
brisunimath is offline  
 
February 21st, 2013, 03:48 PM   #2
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Re: Ring theory problem

Iíll do an example. Suppose you want to find and .

Addition is straightfoward: .

For multiplication, proceed as follows.

1. Multiply the terms out like ordinary polynomials (donít worry about reducing the coefficients mod 2 at this stage):
2. Divide by :
3. Finally reduce the coefficients of the remainder mod 2:

Hence in .
Crimson Sunbird is offline  
February 23rd, 2013, 07:42 AM   #3
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Re: Ring Theory Problem, Help highly appreciated!!

Thankyou very much! That was really helpful
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