Just to give an example, consider the field that you get if you adjoin the cubed root of 3 (call it z) to the rationals. Every number in this field can be written as a + bz + cz² , where a, b, and c, are rational numbers. This makes it a 3-dimensional vector space over Q, even if visually all of these numbers exist one the "one-dimensional" real number line.

Another example would be if z were a primitive third root of unity like e^{2pi / 3 * i}. Again, Q(z) is a field where every element can be written as a + bz + cz² , but now it contains complex numbers. The Eisenstein integers give a nice visual picture of what's going on in this case.