primitive element
Sometimes you build an extension by adjoining several new numbers at once — say both sqrt(2) and sqrt(3). It is a pleasant surprise that you can often achieve the very same field by adjoining just one cleverly chosen number. That single number, which by itself generates the whole extension, is called a primitive element.
Precisely, an element a of an extension L/K is a primitive element if L = K(a) — that is, the entire field L is obtained from K by adjoining this one element a and then closing under the field operations. An extension that has such a single generator is called a simple extension.
The primitive element theorem guarantees these exist remarkably often: every finite separable extension is simple, so it has a primitive element. Since all finite extensions of the rationals (or of any characteristic-zero field) are separable, you can always collapse finitely many adjoined numbers down to one. This is why working with a single generator loses no generality there.
Although Q(sqrt(2), sqrt(3)) seems to need two generators, the single element a = sqrt(2) + sqrt(3) already generates it all: Q(sqrt(2) + sqrt(3)) = Q(sqrt(2), sqrt(3)). So sqrt(2) + sqrt(3) is a primitive element.
sqrt(2) + sqrt(3) generates Q(sqrt(2), sqrt(3)) by itself.