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Section Daily Prep 19
You will become familiar with the basic definitions to begin discussing division and factoring in integral domains. In class, we will introduce Unique Factorization Domains and the relations between these and Principal Ideal Domains.
Objectives: Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State and instantiate the definition of: associate, irreducible, prime
State the following mathematical results: Prime implies Irreducible
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 18, pp. 328, 329
Review Day 17 Notes as needed
Figure 193. Reference Video for Divisibility in Integral Domains
Subsection Important Terms
Definition 194 . Irreducibles and Primes in Domains.
Let \(D\) be an integral domain and \(a,b\in D\text{.}\)
\(a\) and
\(b\) are
associates if
\(a=ub\) for some unit
\(u\in D\)
\(a\) is
irreducible if
\(a\neq 0\) and
\(a\) is not a unit and whenever
\(a=bc\) for
\(b,c\in D\text{,}\) then either
\(b\) or
\(c\) is a unit.
\(a\) is
prime if
\(a\neq 0\) and
\(a\) is not a unit and whenever
\(a|bc\) for
\(b,c\in D\text{,}\) then either
\(a|b\) or
\(a|c\text{.}\)
Theorem 195 . Prime Implies Irreducible.
Let
\(D\) be an integral domain and
\(p\in D\) be prime. Then
\(p\) is irreducible.
