Originally Posted by
YesNo
An algebraic number I think could be defined as the root of a monic polynomial where the coefficients come from the rational numbers, Q. The reason to use a monic polynomial is to avoid having many of these minimal polynomials. For example, x - 7 = 0 gives the same root, 7, as 2x - 14 = 0 does. Since the rational numbers form a field, they have inverses so we can divide by the coefficient of the term with the highest power of x and make it 1. For example, 2x - 3 = 0 can be written as x - (3/2) = 0 and the root 3/2 is an algebraic number. In this case it is a rational number as well.
If it turns out that this minimal, monic polynomial has all integer coefficients, then one defines that root as being not only an algebraic number, but also an algebraic integer no matter what it looks like. That means (1-√5)/2, even though there is a 2 in the denominator, is also an algebraic integer. It is the root of a minimal, monic polynomial with integer coefficients. So it is an algebraic integer.
If we are looking at quadratic algebraic number fields, that is fields where the rationals Q are extended by the square root of a non-square integer, such as √5, then checking whether 5 is congruent to 1 or 3 mod 4 will tell us if the algebraic integers could have a 2 in the denominator or not.