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Section Day 20
This is an outline of the topics we covered in the twentieth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Tuesday 7/28
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
Construct and compute in the field of fractions of an integral domain
State and instantiate the definition of: Euclidean domain
State the following mathematical results: ED Implies PID, D a UFD implies D[x] a UFD.
Definition 203 . Euclidean Domain.
A Euclidean domain (ED) is an integral domain \(D\) with a function \(d: D\setminus\{0\} \to \Z_{\gt 0}\) (the measure ) such that
\(d(a)\leq d(ab)\) for all
\(a,b\in D\setminus\{0\}\)
If \(a,b\in D\text{,}\) \(b\neq 0\text{,}\) then there exist \(q,r\in D\) such that
\begin{equation*}
a=bq+r
\end{equation*}
and either \(r=0\) or \(d(r)\lt d(b)\text{.}\)
Example 204 . Familiar Euclidean Domains.
Theorem 205 . ED Implies PID.
Every Euclidean domain is a principal ideal domain.
Proof.
Theorem 206 . D a UFD Implies D[x] a UFD.
If
\(D\) is a unique factorization domain, then
\(D[x]\) is also a unique factorization domain.
