Skip to main content
Contents
Search Book
Search Results:
No results.
Embed Readability settings Prev Up Next \(\require{mathtools}\require{color} \setcounter{MaxMatrixCols}{15}
\newcommand{\mbu}[1]{\mathbf{\mathrm{#1}}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\GF}{\mathrm{GF}}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\wt}{wt}
\DeclareMathOperator{\im}{im}
\makeatletter
\@ifundefined{char}{
\DeclareMathOperator{\char}{char}
}{}
\makeatother
\DeclareMathOperator{\GRS}{GRS}
\DeclareMathOperator{\RS}{RS}
\DeclareMathOperator{\BCH}{BCH}
\newcommand{\transpose}[1]{#1^{\top}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\bm}{\mathbf{m}}
\newcommand{\bs}{\mathbf{s}}
\newcommand{\be}{\mathbf{e}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\bzero}{\mathbf{0}}
\newcommand{\balpha}{\boldsymbol{\alpha}}
\newcommand{\subgroup}[1]{\langle{#1}\rangle}
\newcommand{\ideal}[1]{\langle{#1}\rangle}
\newcommand{\isom}{\cong}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\cl{cl}
\DeclareMathOperator\Conj{Conj}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\orb{orb}
\DeclareMathOperator\Orb{Orb}
\DeclareMathOperator\Perm{Perm}
\DeclareMathOperator\stab{stab}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\fix{fix}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\Syl{Syl}
\DeclareMathOperator\Sym{Sym}
\DeclareMathOperator\soc{soc}
\DeclareMathOperator{\nil}{nil}
\DeclareMathOperator{\Nil}{Nil}
\DeclareMathOperator\jac{jac}
\DeclareMathOperator\Jac{Jac}
\DeclareMathOperator\Eq{Eq}
\DeclareMathOperator\Hol{Hol}
\DeclareMathOperator\Frac{Frac}
\DeclareMathOperator\Ann{Ann}
\DeclareMathOperator\ev{ev}
\DeclareMathOperator\GL{GL}
\DeclareMathOperator\SL{SL}
\DeclareMathOperator\So{SO}
\DeclareMathOperator\SU{SU}
\DeclareMathOperator\PGL{PGL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\PSU{PSU}
\DeclareMathOperator\PSP{PsP}
\DeclareMathOperator\AGL{AGL}
\DeclareMathOperator\Heis{Heis}
\DeclareMathOperator\Dic{Dic}
\DeclareMathOperator\SA{SA}
\DeclareMathOperator\SD{SD}
\DeclareMathOperator\Fr{Fr}
\DeclareMathOperator\Mod{Mod}
\DeclareMathOperator\DQ{DQ}
\DeclareMathOperator\QD{QD}
\DeclareMathOperator\OD{OD}
\DeclareMathOperator\Cl{Cl}
\DeclareMathOperator\BinTet{2T}
\DeclareMathOperator\BinOct{2O}
\DeclareMathOperator\BinIcos{2I}
\DeclareMathOperator\Aff{Aff}
\DeclareMathOperator\BS{BS}
\newcommand{\normal}{\lhd}
\newcommand{\normaleq}{\unlhd}
\newcommand{\nnormal}{\ntriangleleft}
\newcommand{\nnormaleq}{\ntrianglelefteq}
\def\longto{\longrightarrow}
\def\into{\hookrightarrow}
\def\longinto{\longhookrightarrow}
\def\onto{\twoheadrightarrow}
\DeclareRobustCommand\longonto{\relbar\joinrel\twoheadrightarrow}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator\Ker{Ker}
\DeclareMathOperator{\Id}{Id}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\definecolor{xRed}{RGB}{229, 31, 58}
\definecolor{xBlue}{RGB}{68, 119, 170}
\definecolor{xGreen}{RGB}{33, 135, 51}
\definecolor{xPurple}{RGB}{170, 51, 119}
\definecolor{xOrange}{RGB}{ 197, 83, 17}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\newcommand{\fillinmath}[1]{\mathchoice{\underline{\displaystyle \phantom{\ \,#1\ \,}}}{\underline{\textstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptstyle \phantom{\ \,#1\ \,}}}{\underline{\scriptscriptstyle\phantom{\ \,#1\ \,}}}}
\)
Section Daily Prep 20
Today we will investigate the field of fractions of an integral domain, which is constructed in a manner similar to the construction of the rational numbers from the integers. We will also investigate the relationship between Euclidean domains, principal ideal domains, and unique factorization 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.
Construct and compute in the field of fractions of an integral domain
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 15, pp. 290-292
Figure 201. Reference Video for Fields of Fractions
Subsection Important Terms
Theorem 202 . Field of Fractions.
Let
\(D\) be an integral domain. Then there exists a field
\(F\) (called the field of fractions of
\(D\) ) that contains a subring isomorphic to
\(D\text{.}\)
