Originally Posted by

**YesNo**
I think it is only mod 4 that is of interest here, not mod 3 or mod 5.

For quadratic algebraic number fields, that is fields where the numbers are roots of quadratic equations like ax^{2}+bx+c=0, the way to tell what algebraic numbers are integers is given by the rule that if we extend Q by the square root of a negative integer, then if that negative integer, -t, is congruent to 3 mod 4 then the algebraic integers are what we would expect them to be, numbers like a+b*sqrt(-t). If -t is congruent to 1 mod 4 then we have to also include a/2+b/2*sqrt(-t) as integers. This comes from using the quadratic formula to find x = (-b+-sqrt(b^{2}-4ac)/2a. That 2 in the denominator does not cancel out in this case. The a = 1 because that is required for an algebraic number to be an algebraic integer. It has to be monic.

Don't worry about being out of reach. If I can't explain it, then I don't really understand it well enough and I don't understand this myself all that well. Also, I might have some of this wrong.