Primes in Quadratic Fields
I was examining a case described in my book about primes in Quadratic Fields. The case describe examines positive prime numbers named "d" that are in Z and meet the condition that the quadratic integers in Q[Sqrt(d)] are in the form of a+b*Sqrt(d) , where of course a and b must be rational.
I need a place to start with this before I can continue. Can someone give me the first 3 or four of such primes? (Call these solutions - A)
What if they were required to be in the form of (a+b*Sqrt(d))/2 and 'a' and 'b' had to be both even or both odd? Do you just divide what we called Solutions-A by 2 as long as the numbers are still rational? I don't think that would work because I don't know of many primes that function in this way. What would the first 3 or 4 such primes of this form be? (Call these solutions - B).
Lastly, how would this change if we replaced Sqrt(d) with Sqrt(-d) for all mentioning of d above? As in, how would solutions A and solutions B differ? What would they be?
Thank you all, I really appreciate the help!