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3.6. Rational Functions 107

(v) ala2a3 .--ak^3 nk(ula2. ..ak) (mod m).

Deduce that n4(*) 3 1 (mod m), and obtain the Little Fermat Theorem

as formulated in Exploration E.32 as a corollary.

3.6 Rational Functions

For any field F, the set of polynomials F[t] with the usual addition and
multiplication is a commutative ring like Z. Just as Z can be embedded in
the field Q, F[t] is contained in the field of rational functions. These are
expressions of the form p(t)/q(t) where p and q are polynomials over F and
q(t) is not the zero polynomial. Addition, subtraction, multiplication and
division of rational functions are carried out as for rational numbers. An
important difference between polynomials and rational functions is that it
is not always possible to evaluate a rational function at every point of its
underlying field F. A rational function p(t)/q(t) assumes the value p(c)/q(c)
when t = c, except when c is a zero of q, in which case the value is left
undefined. For example, (3t+2)/(t2-4) is a rational function over Q whose
value is undefined at 2 and -2.

Exercises

1. Show that every rational function f(t) can be written in the form
   p(t) + g(t), where p(t) is a polynomial (possibly 0) and g(t) is a
   rational function the degree of whose numerator is smaller than the
   degree of the denominator.


2. Show that the rational function
   at + b
   (t - m)(t - n)
   can be written in the form
   A B
   -+-
   t-m t-n
   for some constants A and B determined by

A(t - n) + B(t - m) = at + b.

Just.ify the assertion that the appropriate values of A and B can be

found by making the substitutions t = m and t = n. Find A and B

by this method and check that the values obtained are correct.

3. Write the rational function
   t + 14
   (t - 1)(t + 4)