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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
.
If *n* = 4*k*+1 is a prime, then -1is a non-residue, and we can find its square root.
If
*n*=*n*_{1} *n*_{2} ... is composite, then -1 should be a non-residue
for all *n*_{i}.
As a consequence *n* has two possible forms, *n*=4*k*+1 or
*n*=8*k*+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*)-*x*^{2}-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

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.

** Next:** Square roots of integers
** Up:** Modular mappings
** Previous:** Arbitrary powers
*Gaston Gonnet*

*1999-07-04*