Let f: R S be a ring homomorphism.
(a) Prove that kernel(f) is an ideal of R.
(b) Prove that if f is surjective, then image(f) is an ideal of S.
(10) Let
Z(√3)= {a+b√3: ab € Z}.
Define
N(a+b√3)=a²-3b²
(a) Let 5+2√3 and v=7-3√3
Compute u + vand ue.
(b) Let
x=a+b√3 and y=c+ √d
Prove that N(xy) = N(x)N(y).