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30 CHAPTER 2. POLYNOMIALS
7. a ∗ 1 = a;
8. a ∗ (b + c) = (a ∗ b) + (a ∗ c);
9. m + n = m ⊕ n;
10. m ∗ n = m ⊗ n;
where we have used ⊕ and ⊗ to denote the operations of addition and multiplication
on coefficients, which are assumed to be given to us.
The reader can think of the coefficients as being numbers, though they need not
be, and may include other indeterminates that are not the “certain indeterminates”
of the definition. However, we will use the usual ‘shorthand’ notation of
2 for 1 ⊕ 1 etc. The associative laws (2 and 6 above) mean that addition and
multiplication can be regarded as n-ary operations. A particular consequence
of these rules is
8’ m ∗ a + n ∗ a = (m ⊕ n) ∗ a
which we can think of as ‘collection of like terms’.
Proposition 6 Polynomials over a ring form a ring themselves.
If it is the case that a polynomial is only zero if it can be deduced to be zero by
rules 1–10 above, and the properties of ⊕ and ⊗, then we say that we have a free
polynomial algebra. Free algebras are common, but by no means the only one
encountered in computer algebra. For examples, trigonometry is often encoded
by regarding sin θ and cos θ as indeterminates, but subject to sin 2 θ + cos 2 θ = 1
[Sto77].
Notice what we have not mentioned: division and exponentiation.
Definition 20 ((Exact) Division) If a = b ∗ c, then we say that b divides a,
and we write b = a c .
Note that, for the moment, division is only defined in this context. We note
that, if c is not a zero-divisor, b is unique.
Definition 21 (Exponentiation) If n is a natural number and a is a polynomial,
then we define a n inductively by:
• a 0 = 1;
• a n+1 = a ∗ a n .
Notation 10 If K is a set of coefficients, and V a set of variables, we write
K[V ] for the set of polynomials with coefficients in K and variables in V . We
write K[x] instead of K[{x}] etc.