Exponentials of complex numbers pose a more immediate problem.
We have seen that to represent *i*, *n* has to be either of the
form 4*k*+1 or 8*k*+2.
In both cases, is a multiple of 4 and it is not
possible to represent
.
Since requires *i* to be represented in the base and in the
exponent, we need to resolve this problem if want to compute
signatures of trigonometric functions.

To represent *i* and the square root of other non-residues we
will use one field extension.
We will call this extension , and since it has no
representation (mod *n*) we will have to operate with it in
symbolic form.
Without loss of generality, we will assume that it satisfies
.
So when this is needed, our signatures will be of the form
and called -forms.
(It is easy to see that the signatures of the square roots
of non-residues are of the form , so as a side effect
we can represent the square roots of all the integers.)
Arithmetic is done as with polynomials in and
every time that we generate we substitute it for -1.
Inverses and divisions are done by multiplying the numerator and
denominator by the conjugate with respect to .
The only problem to resolve is how to compute exponentials of
-forms, and here is where we will introduce signatures
of trigonometrics.

Let
, where *r* and are relatively
prime, neither *r* nor
are *small*
and *r* is otherwise randomly chosen.
With such an *r*, *T* is also a primitive root (mod *n*).
Since neither *r* nor 1/*r* are small, there is little chance, *O*(1/*n*),
of polynomials in *S*(*e*)^{x} being confused with polynomials in *T*^{x}.
In what follows, *n* will be chosen so that *i* can be represented.
The signatures of the trigonometric functions will be
computed as follows.

The rules for computing exponentials and trigonometrics with -forms are:

etc. In simpler terms, the symbol is assigned an arbitrary value and forced to respect the only simplification restriction which it has to follow: .

It is not difficult to verify that with these definitions, , , , , etc.