A.4 Fields and rings
One interesting observation from the examples in the previous section is that each of the sets Z_{p}, R, Q, and C contains two different abelian group structures: the set itself under addition and the set of nonzero elements under multiplication. Structures satisfying this property together with an axiom about multiplication ``distributing'' over addition are called fields.
Formally, a field consists of a set F together with two operations + : F ×F ® F and ·: F ×F ® F called addition and multiplication, respectively, such that the following axioms are satisfied.
- F forms an abelian group under addition.
- F \{0} forms an abelian group under multiplication, where 0 is the identity in the additive abelian group áF, + ñ.
- Multiplication distributes over addition, that is, a ·(b+c) = a ·b + a ·c.
For an integer n and a field element x, n ·x denotes the element obtained by adding x to itself n times; for example, 3·x = x+x+x. The characteristic of a field is the smallest positive integer p such that p ·1 = 0. If no such p exists, then the characteristic of the field is defined to be 0. The characteristic of a field is either a prime number or 0. If the characteristic of a field is 0, then the field is infinite. However, a field with nonzero characteristic might be either finite or infinite.
Examples.
The fields Q, R, and C of rational, real, and complex numbers, respectively, are fields of characteristic 0. The finite field Z_{p} is a field of characteristic p.
The number of elements in a finite field must be a power of a prime number. A classification theorem of the finite fields states that there is exactly one finite field (up to isomorphism; see [Fra98]) of size q for each prime power number q. Thus it makes sense talking about the field with q elements, which is traditionally denoted GF(q) (GF = Galois Field) or F_{q}.
A ring R satisfies axioms (1) and (3), but instead of (2), multiplication in R is only required to be associative. If multiplication is commutative, then the ring is commutative. A nonzero element a in a ring is a zero divisor if there is a nonzero element b such that ab = 0. There are two main classes of commutative rings:
- Rings with no zero divisors. All fields and the ring Z of integers are of this kind.
- Rings with zero divisors. The ring Z_{n} contains zero divisors if and only if n is composite.
A polynomial in a ring R is a function f : R ® R of the form
f(x) = a_{0} + a_{1} x + a_{2} x^{2} + ¼+ a_{n} x^{n}, |
where a_{0}, ¼, a_{n} are elements in the ring. A root of a polynomial is an element r such that f(r) = 0.