Finite Cyclic Groups and Conditions on n for n to be Cyclic
In Module II we determined that is a finite group of order (n) and
that if a then ord (a)
(n). Of course, if ord (a) = (n) then
= 1, a, a 2 ,., a (n)-1
and we say that is cyc
Elements of Number Theory We begin with a basic study of the multiplicative structure of
and we introduce the Euler phi function, which plays an important role in the study of residue
Definition 1 If a, b we say a divides b, denoted b
Quadratic Residues: The Legendre and Jacobi Symbols The apparent difficulty of determining
quadratic residues (the Quadratic Residuosity Problem) is the basis for believing the
Goldwassor-Micali probabilistic public-key encryption scheme to be
MORE ALGEBRA POLYNOMIAL RINGS
To prepare for a study of finite fields (considered in Module VI) we begin with the notion of a ring of
polynomials with coefficients from an arbitrary field.
Definition 1. Let F be any field and x be a variable (s
Finite Fields This module deals with the existence and the structure of finite fields.
Remarks 1. F x (f(x) is a finite field when F is finite and f(x) is irreducible.
Proposition 1. If F is a finite field then a smallest number p, which is ne
More Number Theory and Some Algebra; n ( mod n)
As we already know (, +, o has "algebraic" structure.
Indeed. both + and o are binary operations on that are associative and
commutative; 0 is the identity of (, +), 1 is the identity of (,o an