Answer to Question #47367 in Abstract Algebra for ayush kumar

Question

a) Let R={a+b(root under 5 I a,bEZ} and d:R--->Z be a function defined by d{a +b(root under 5)} = a2 — 5b2.
(i) Check whether d is a ring homomorphism. If it is, find its kernel.

(ii) Show that, if a(alfa) E R is a unit, d{a(alfa)} = -±1.

(b) Check whether {(1, 1), (2, 2), (1, 2), (2, 3), (3, 3), (2, 1), (3, 2)} is an equivalence relation on {1, 2, 3} or not. Give reasons for your answer.

(c) Let Q be the field of rational numbers .
Define a e b= a+ b, a 0 b= 2.b. Check whether (Q, + , .) is a commutative ring with identity.