Prime Numbers

2.2 Polynomial arithmetic 89
is used for repeated CRT calculations, one can perform [Precomputation] for
Algorithm 2.1.7 just once, store an appropriate set of r − 1 integers, and allow
efficient reentry.
In Section 9.5.9 we describe a CRT reconstruction algorithm that not only
takes advantage of preconditioning, but of fast methods to multiply integers.
2.2 Polynomial arithmetic
Many of the algorithms for modular arithmetic have almost perfect analogues
in the polynomial arena.
2.2.1 Greatest common divisor for polynomials
We next give algorithms for polynomials analogous to the Euclid forms in
Section 2.1.1 for integer gcd and inverse. When we talk about polynomials,
thefirstissueiswherethecoefficientscomefrom.Wemaybedealingwith
Q[x], the polynomials with rational coefficients, or Zp[x], polynomials with
coefficients in the finite field Zp. Or from some other field. We may also be
dealing with polynomials with coefficients drawn from a ring that is not a
field,aswedowhenweconsiderZ[x] orZn[x] withn not a prime.
Because of the ambiguity of the arena in which we are to work, perhaps
it is better to go back to first principles and begin with the more primitive
concept of divide with remainder. If we are dealing with polynomials in F [x],
where F is a field, there is a division theorem completely analogous to the
situation with ordinary integers. Namely, if f(x),g(x) areinF [x] withf not
the zero polynomial, then there are (unique) polynomials q(x),r(x) inF [x]
with
g(x) =q(x)f(x)+r(x) and either r(x) =0or degr(x) < deg f(x). (2.4)
Moreover, we can use the “grammar-school” method of building up the
quotient q(x) term by term to find q(x) and r(x). Thinking about this
method, one sees that the only special property of fields that is used that
is not enjoyed by a general commutative ring is that the leading coefficient
of the divisor polynomial f(x) is invertible. So if we are in the more general
case of polynomials in R[x] whereR is a commutative ring with identity, we
can perform a divide with remainder if the leading coefficient of the divisor
polynomial is a unit, that is, it has a multiplicative inverse in the ring.
For example, say we wish to divide 3x +2intox 2 in the polynomial ring
Z10[x]. The inverse of 3 in Z10 (which can be found by Algorithm 2.1.4) is 7.
We get the quotient 7x + 2 and remainder 6.
In sum, if f(x),g(x) areinR[x], where R is a commutative ring with
identity and the leading coefficient of f is a unit in R, then there are unique
polynomials q(x),r(x) inR[x] such that (2.4) holds. We use the notation
r(x) = g(x) modf(x). For much more on polynomial remaindering, see
Section 9.6.2.