Factorization

Terminology

Property	Statement	Ideals
is a unit	has mult. inverse in
divides	for
proper divisor of	where not unit
and are associates	where unit
is irreducible	nonunit, no prop. div.
is prime	implies or	if

Euclidean Domains

Examples: , ,

For a domain , the existence of a size function implies is a Euclidean domain, if , , such that where or

Principal Ideal Domain

Examples:

An integral domain where every ideal is principal is a principal ideal domain.

Let an integral domain not both zero. A greatest common divisor of and is an element where and . If , , then .

If the is a unit, then and are relatively prime. This will always exist in a PID.

In a PID, an element is irreducible IFF it is a prime element. Furthermore, the maximal ideals are principal ideals generated by irreducible elements.

Unique Factorization Domain

Examples: ,

A domain is a unique factorization domain if for all elements, factorization terminates and is unique up to associates.

This is equivalent to stating that does not have an infinite strictly increasing chain of principal ideals.

In an integral domain, if every irreducible element is a prime element, then the domain is a UFD

Any pair of elements has a .

Integral Domain

Examples:

An integral domain or domain is a ring where there are no zero divisors, that is, implies or .

In an integral domain, prime elements are irreducible.

Gauss Lemma

For a prime integer , denote . iff divides the coefficients of , i.e. in .

is primitive when is 1 and

Integers are prime elements in , so iff or . Moreover (Gauss Lemma) the product of primitive polynomials is primitive.

Factoring Integer Polynomials

Let be an integer polynomial, and a prime that does not divide leading coefficient of . If is irreducible in , then is irreducible in .

Eisenstein Criterion Let be an integer polynomial and a prime integer. If does not divide , p divides , and does not divide , then is irreducible in