Thread: Determine the number of Elements

HELP!!!!
Let A denote the ideal of Z3[x] generated by the polynomial x^2+x+1, that is,

A={f(x)(x^2+x+1)|f(x) is an element of Z3[x]},

Determine the number of elements in the factor ring Z3[x]/A.

Originally Posted by fourth

HELP!!!!Let A denote the ideal of Z3[x] generated by the polynomial x^2+x+1, that is, A={f(x)(x^2+x+1)|f(x) is an element of Z3[x]}. Determine the number of elements in the factor ring Z3[x]/A.

Hint: $\displaystyle p(x)+A=q(x)+A\Leftrightarrow p(x)-q(x)\in A$ . Now use the euclidean division to prove that every element $\displaystyle \mathbb{Z}_3/A$ can be uniquely expressed by $\displaystyle p_1(x)+A$ with $\displaystyle \textrm{deg}(p_1(x))<2$ .


Edited: Of course I meant $\displaystyle \mathbb{Z}_3[x]/A$ instead of $\displaystyle \mathbb{Z}_3/A$ .