Page 27 - Hardware Implementation of Finite-Field Arithmetic

10 Cha pte r O n e

1.2.3 Fields

Definitions 1.14 A field (F, +, ) consists of a set F with two binary
*
operations + and , with an additive identity element 0 and a multiplicative

*

identity element 1 satisfying the following axioms:

1. (F, +, ) is a commutative ring.
*
2. All nonzero elements of F have a multiplicative inverse.
Definition 1.15 The characteristic of a field is the least positive integer
=
m such that ∑ m = i 1 10. Otherwise, the characteristic of a field is 0 if
11++ ... + 1 (m times) is never equal to 0 for any m > 0.

Examples 1.8
• The real numbers R form a field of characteristic 0 under the
usual operations.
• The set of integers Z with the usual operations of addition (+)
and multiplication (·) is not a field, because the only nonzero
elements with multiplicative inverses are 1 and −1.
• The set Z with the usual operations of addition and
p
multiplication modulo p is a field if and only if p is a prime. If
p is prime, then Z has characteristic p.
p
• Consider the field Z . The tables for the addition and multi-
5
plication operations modulo 5 are as follows (Table 1.1):

+ 0 1 2 3 4 ⋅ 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1

TABLE 1.1 Addition and Multiplication over Z
5

Definitions 1.16
1. A subset E of a field F is a subfield of F if E is itself a field with
respect to the operations of F. In such a case, F is said to be an
extension field of E. If E ≠ F, we say that E is a proper subfield of F.
2. A field containing no proper subfields is called a prime field .