Factorization in integral domains

Factorization in Integral Domains II

Theorem 1.1 (Rational roots test). Let f = anx n + · · · + a0 ∈ Z[x] be a polynomial of degree n ≥ 1 with integer coefficients and nonzero constant term a0, and let p/q ∈ Q be a rational root of f such that the fraction p/q is in lowest terms, i.e. gcd(p, q) = 1. Then p divides the constant term a0 and q divides the leading coefficient an. In particular, if f is monic, then a rational root of f...

Factorization in Integral Domains I

Definition 1.1. For r, s ∈ R, we say that r divides s (written r|s) if there exists a t ∈ R such that s = tr. An element u ∈ R is a unit if it has a multiplicative inverse, i.e. if there exists an element v ∈ R such that uv = 1. The (multiplicative) group of units is denoted R∗. If r, s ∈ R, then r and s are associates if there exists a unit u ∈ R∗ such that r = us. In this case, s = u−1r, and ...

Unique Factorization in Integral Domains

Throughout R is an integral domain unless otherwise specified. Let A and B be sets. We use the notation A ⊆ B to indicate that A is a subset of B and we use the notation A ⊂ B to mean that A is a proper subset of B. The group of elements in R which have a multiplicative inverse (the group of units of R) is denoted R×. Since R has no zero divisors cancellation holds. If a, b, c ∈ R and a 6= 0 th...

Unique Factorization Monoids and Domains

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e...

Lectures On Unique Factorization Domains