How do you find the extension field when E=Z3[x]/x^2+1 of Z3, list the nine elem

Question

How do you find the extension field when E=Z3[x]/x^2+1 of Z3, list the nine elements. I like to see the work for at least three elements. Please and thank you. My prof uses Ex. Alpha squared + alpha + 1= 0 then Alpha squared=(-1)alpha +(-1)=alpha + 1 for z2 How do you find the extension field when E=Z3[x]/x^2+1 of Z3, list the nine elements. I like to see the work for at least three elements. Please and thank you. My prof uses Ex. Alpha squared + alpha + 1= 0 then Alpha squared=(-1)alpha +(-1)=alpha + 1 for z2

Explanation / Answer

any polynomial p=(x^2+1)q+r in Z3[x], where degree r is less than 2,

so modulo x^2+1, q=r

So we need to find the polynomials of degree less than 2, which are

0,1,2

x,x+1,x+2,2x,2x+1,2x+2

How many are? qed

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