Question

We have to determine whether the set of all integers of the form m+(n(sqrt(3))) is a ring w.r.t to addition and multiplication.

Is there any identity element e which belongs to this set, under addition such that x + e = x = e+x for all x belongs to above set?

Since e should also be of the form m+(n(sqrt(3))), and e should be 0, which is, e = 0+(0*sqrt(3))), can we take this belongs to the set since both m=n=0?

Answer 1

We have to determine whether the set of all real numbers of the form m+(n(sqrt(3))) , where m,n are integers, is a ring w.r.t to addition and multiplication*

Answer 2

looks like e=0+0√3 works for addition
for multiplication, I get
(a+b√3) = (-m+n√3)/(3n^2-m^2)
So, it looks like there is no multiplicative identity if 3n^2 = m^2
Luckily, there are no integers where this is true.
so, for example, (2+5√3)*(-2/71 + 5/71 √3) = 1