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 Fields of Fractions and Euclidean Domains
Worksheet Part 1: Fields of Fractions
In this activity, you’ll practice constructing the field of fractions of an integral domain and showing that it is isomorphic to a known field.
1. Let
\(\Z[i]=\{a+bi\mid a,b\in \Z\}\text{.}\) Show that the field of fractions
\(F\) of
\(\Z[i]\) is ring-isomorphic to
\(\Q[i]=\{r+si\mid r,s \in \Q\}\text{.}\) It wil lbe helpful to show that any fraction
\((a+bi)/(c+di)\in F\setminus\{0\}\) is equivalent to a fraction of the form
\((x+yi)/z\) with
\(x,y,z\in \Z\) and
\(z\gt 0\text{,}\) which you can then use to define a map from
\(F\) to
\(\Q[i]\text{.}\) You will need to show that your is a well-defined ring homomorphism that is 1-to-1 and onto.
Worksheet Part 2: \(\Z[i]\) is a Euclidean Domain
In this activity, you’ll practice showing that a ring is a Euclidean domain.
1.
In this problem, you will show that the set of Gaussian integers
\(\Z[i]\) is a Euclidean domain. You may assume that the norm function
\(d(a+bi)=a^2+b^2\) satisfies the norm properties without proof.
(a) Show that
\(d(x)\leq d(xy)\) for
\(x,y \in \Z[i]\text{.}\)
(b)
Show that if \(xy^{-1}=s+ti \in \Q[i]\) (the field of fractions of \(\Z[i]\) ), then if \(m\) and \(n\) are the nearest integers to \(s\) and \(t\) respectively we have
\begin{equation*}
x=(m+ni)y+[(s-m)+(t-n)i]y.
\end{equation*}
(c)
Show that the division condition of a Euclidean domain is satisfied with
\begin{equation*}
q=m+ni, \qquad r=[(s-m)+(t-n)i]y,
\end{equation*}
using the fact that we must have \(|m-s|,|n-t|\leq 1/2\text{.}\)
(d) Apply the above to find the quotient and remainder upon dividing
\(3-4i\) by
\(2+5i\text{.}\)
