Example

Let a be the (positive) square root of 2. Thus, a = 2. Since a² = 2, it follows that for every polynomial f the number f(a) is of the form c + da. So Q(a) consists of the quotients

(c + da)/(g + ha),

with c, d, g, h Q. These expressions can be simplified even further: multiply numerator and denominator by g - ha to conclude that

Q(a) = {c + da | c, d Q}.

In other terms,

Q(2) = Q + Q2.