Each of the quotient rings /I on the left is isomorphic to one of the rings S on the right. Match each ring with its isomorphic partner, and prove that they really are isomorphic by constructing a surjective ring homomorphism : R S with kernel I. R/I S A) Z[X]/(8.12, X) 1. Z 2. Q 3. R B) Q[X]/(X? -2) 1. C 5. Fy . 6. Fy C) RIX]/(X - v2) 7, 8. Zg D) RIX]/(X2+ X +2) 9. {a+bv/2]a,beQ} 10. {(g i) |a,be_*'4} E) RIX]/X5) 11. {( a z) la,b Fy F) Z,[X)/(X* + X% + 1) 2 {(Z i) a,bER}} In examples 5, 6, 10, and 11 the symbols F; and Fs denote a field of 4 and a field of 8 elements respectively. In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices. Some methods you might use to try and match up R/I and S: (i) guess, (i) think about representatives in the quotient ring, and what the multiplication rules are and try and match that up with something in the S column, (i) try and make up a homomorphism from R to somewhere that would have elements of the ideal I in the kernel