I hope this helps: in order to prove a ring is field we have to prove that the ring has multiplicative inverse, that is it is a unit.

q(rood (d)) is the elements that contains Q and rood (d)

ie: all the elements of the form a + b*sqrt(d) .now you have to see if there exist another element in this form such that (a+ b*root(d))* (c+ d*root(d))=1

I recomend that you divide 1/(c+d*root(d)) and see if you can find an element in Q(root(d))

Hint: use the fact (a-b)*(a+b)=(a^2)-(b^2)