27-11-2012, 02:31 PM
Introduction to Algebra
Algebra[.ppt (Size: 296 KB / Downloads: 22)
Integral Domains
A Zero-Divisor is a nonzero element a є R, R is a commutative ring, such that there is a nonzero element b є R with ab = 0.
A Unity in a ring is a nonzero element that is the identity under multiplication.
An Integral Domain is a commutative Ring with unity and no zero-divisors.
Rings
A ``ring'' is a set of elements having two operations, usually called addition and multiplication, which behaves in many ways like the integers.
You can add and multiply elements in a ring but you can't (usually) divide them and get another element in the ring.
Fields
A ``field'' is a set of elements having two operations, usually called addition and multiplication, which behaves in many ways like the rational numbers.
You can add and multiply elements in a field and you can divide one element by any non-zero element to get another element in the field.
More Properties of GF(pn)
It can be shown that for each positive integer n there exists an irreducible polynomial of degree n over GF(p) for any p.
It can be shown that for each divisor m of n, GF(pn) has a unique subfield of order pm. Moreover, these are the only subfields of GF(pn).