3.8 Doubling down

So far, we’ve seen finite and infinite groups, rings, and fields, some of which are extensions of others. In this section, we look at combining them.

Consider the collection of all pairs of integers (a, b), where we define addition and multiplication component-wise.

This is a ring, denoted Z², but it is not an integral domain. (1, 0) × (0, 1) = (0, 0), but neither of the factors is 0.

For the same reason, neither Q² nor R² can be an integral domain. In particular, they are not fields with these operations.

Let’s change the definitions for R² so that 1 = (1, 0) and multiplication is

For (a, b) ≠ 0, we define

With these unusual definitions for multiplication and inversion, we not only have an integral domain, we have a field, which we examine in the next section.