An algebraic entity consisting of a set of elements and two binary
associative operations on those elements, usually called addition and
multiplication. If the multiplication operation is not commutative,
we call it a skew field or division ring. If there are non-zero elements
which don't have multiplicative inverses, we have a ring.
In Combinatorics the fields of greatest interest are finite. A finite
field must have size equal to a power of a prime, and are usually denoted
GF(p) (for Galois Field). The finite field with two elements, GF(2), is
of particular importance. GF(2) consists of the two elements 0 and 1, connected
by the operations + and * as in the following tables:
+ | 0 1
---+--------
0 | 0 1
|
1 | 1 0
* | 0 1
---+--------
0 | 0 0
|
1 | 0 1
--back to combinatorics--