= Ch 4: Finite Fields =
Groups, Rings and Fields
A group is sometimes noted latex2($\{G, \cdot\}$). Where G is the set of elements and the dot is a binary operator. A group can also be an abelian group, ring, commutative ring, integral domain or field, each of which has additional restrictions. Here we give those restrictions:
Group
- Closure under addition
- Associativity of addition
- Additive identity
- Additive inverse
Abelian Group adds
- Commutativity of addition
Ring adds
- Closure under multiplication
- Associativity of multiplication
- Distributive laws
Commutative ring adds
- Commutativitiy of multiplication
Integral Domain
- Multiplicative Identity
- No zero divisors (note that by adding multiplication in a finite group, we automatically define the ability to divide because we can define division of a/b=c as b*c=a. What we don't know is if c exists. To get around the divide by zero problem we restrict the notion of division to exclude a divisor of zero.
Field adds
- Multiplicative inverse.
Modular Arithmetic
Euclid's Algorithm
Finite Fields of the form GF(p)
Polynomial Arithmetic