Question

Let R = Z[ √2 ] and M = {a + b √2 ∈ 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 (a^2 + b^2), 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.