I am going to cite the book "An Introduction to Number Theory" (Harold M. Stark) which I recently picked up. In it, they say the following:

Let d be a fixed rational number which is not the square of a rational number. We let

denote the set of numbers

where

and

are arbitrary rational numbers. We call

a quadratic field and if

, it is a real quadratic field but if

, it is a complex or imaginary quadratic field. As an example

is a member of

.

So using this, wouldn't one write a proof showing that there exists an infinite number of primes of the form

? From my mind, d has to be a perfect even square number,

and/or

? That makes sense to me however I'm not sure if its true.

As an example:

Prime

Not prime (not an even square)

Prime

Not prime (not an even square)

Prime

If this is wrong, someone feel free to correct me.