Complex numbers

Working with complex numbers reduces to being able to compute the signature of i, the imaginary unit. S_n(i) should satisfy the basic relation of S_n(i)²+1=0. This means that $S_n(i) = \sqrt{-1} \bmod n$. If n = 4k+1 is a prime, then -1is a non-residue, and we can find its square root. If n=n₁ n₂ ... is composite, then -1 should be a non-residue for all n_i. As a consequence n has two possible forms, n=4k+1 or n=8k+2. Once that we have chosen n so that S_n(i) has a representation, we still have two choices (plus and minus). Any choice works provided that we use it consistently. This choice is not yet a problem, but will become one once that we introduce algebraic numbers in general. This will be discussed later.

Suppose we want to test whether (x+i)(x-i)-x²-1 is identically 0. We can work with n=13, where S(i)=5 is a possible choice. Selecting S(x)=2 as before we obtain

$$7 \times (-3) - 4 - 1 = -26 = 0 \pmod{13}$$

When we are forced to compute signatures in a field or congruence which does not allow the representation of i, we will use field extensions. This will be described for the signatures of e^x and trigonometric functions.