Answer to Question #66828 in Abstract Algebra for Pushpa

Question

Let R= Z[root 2] and M={a+b root2€ R|5|a and 5|b}.
i) Show that M is an ideal of R
ii) Show that if 5 |a or 5|b , then 5|(a2 + b2), for a,b € Z
iii) Hence show that if N is an ideal of R properly containing M, then N= R
iv) Show that R/M is a field , and give two distinct non- zero elements of this field