1. We have shown that Z3[$]/ amp;lt; (1.352 | bar; + c amp;gt; is a eld if and only if (1562 | bx + c is irreducible. (a) When do two quadratic

1. We have shown that Z3[x]/ < ax2 +bx+c > is a field if and only if ax2 + bx + c is irreducible.

(a) When do two quadratic polynomials ax2 + bx + c generate the same ideal in Z3[x]?

(b) How many maximal ideals generated by quadratic polynomials does Z3[x] contain?

2. LetF1 =Z3[x]/<x2+2x+1>andletF2 =Z3[x]/<x2+x+2>

1. (a) How many elements does F1 have? F2?
2. (b) Are F1 and F2 isomorphic? if isomorphic, show it.