\def\real{{\bf R}}
\def\comp{{\bf C}}
\def\quat{{\bf H}}
\def\int{{\bf Z}}
\magnification=\magstephalf
\def\gap{\hskip .3cm}
\nopagenumbers

{\bf \centerline{Math 222  Exam 2: Friday, March 21, 1997; 1:55--2:45pm
 }} \bigskip\noindent{\bf Instructions.} 
 Use
your time wisely; the four questions are  not of equal value.
  Calculators may be used if desired.
 Throughout the exam, $\int$ denotes the integers and
 $\int_m$ denotes the integers modulo $m$.
\bigskip

 \noindent{\bf 1.} {\sl (30 points)}\smallskip
(a) Express the following complex numbers in the form $a+bi$ for real
numbers $a$, $b$;\par\gap (i)\gap $(2+i)-(3-2i)$\gap (ii)\gap $(2+i)(3-2i)$\gap
(iii)\gap $\overline{3-2i}$\gap (iv)\gap  $\vert 3-2i\vert$\gap (v)\gap
$(3+2i)^{-1}$. \smallskip  \smallskip (b)
If the polar form of a complex number $z$ is $z=r(\cos \theta +i\sin \theta)$
and $m$ is a positive integer, express $z^m$ in polar form. 
 Find all the complex numbers $z$ such that $z^4=-4$ and express each of  them
in the form $a+bi$ with $a$, $b$ real numbers.  
\smallskip (c) Calculate the following quaternions, expressing
the answers in the form $a+bi+cj+dk$ with $a$, $b$, $c$, $d$ real numbers;
\par\gap (i)\gap  $(2+i-j+3k)+(3+2i-2j-2k)$ \gap (ii) \gap $(1+i-j)(i+j+k)$
\gap (iii) $(2-i+2j-2k)^{-1}$.\bigskip


\noindent{\bf 2.}  {\sl (30 points)} Let $R$ denote the ring of polynomials
$\int_3[x]$ in the variable $x$ over $\int_3$ and $F$ denote the
 field of fractions of $R$.\smallskip (a) Let $f=\overline{2}+x+\overline{2}x^2$
and $g=\overline {2}+x$ Express the 
polynomials \gap (i)\gap $f-g^2$ and \gap  (ii)\gap $g^3$  in the form
$a_0+a_1x+\ldots+ a_nx^n$ with each $a_i\in \int_3$ one of $\overline 0$,
$\overline 1$ or $\overline 2$.

\smallskip (b) In $F$, express
$x^{-1}+(x+\overline{1})^{-1}+(x+\overline{2})^{-1}$ in the form $h\over k$ 
with $h$, $k\in R$.  \smallskip (c) Prove that for any $h\in R$, $h+h+h=0$
 (hint;
first explain why $h+h+h=(\overline {1}+\overline{1}+\overline{1})h$).
Is $h+h+h=0$ true for any $h\in F$? Give reasons for your answer.\smallskip
(d) Let $S$ denote the subring of $R$ generated by $\int_3$ and $x^2+\overline
1$. Which of the following elements of $R$ are in $S$ \par \gap (i) \gap $x^2$
 \gap(ii)\gap  $x^4+\overline{2}x^2+\overline{2} $\gap (iii)\gap 
$x^2+x+\overline{2}$ \gap?\par Under what  conditions on  $a_0,a_1,\ldots, a_n\in
\int_3$ is $a_0+a_1x+\ldots +a_nx^n$ an element of $S$? \bigskip 

\noindent{\bf 3.}  {\sl (25 points)}\smallskip (a) Prove carefully that the
relation $\sim$ on $\int$ defined by $a\sim b$ if $\vert a\vert=\vert
b\vert$, where  $\vert\ \vert$ denotes absolute value, is an equivalence
relation. List the elements of each of the equivalence classes $[0]$, $[1]$ and
$[2]$.
 \smallskip (b) Let $Y=\int/\negthinspace\sim$ denote the set of equivalence
classes for the equivalence relation in (b). Prove that there is a well-defined
function $f\colon Y\rightarrow \int$ such that $f([x])=x^2$ for any $x\in \int$,
but there is no function $g\colon Y\rightarrow \int$ with
$g([x])=x^3$. Calculate $f([0])$ and $f([2])$. For which positive integers
 $m$ is
there a well-defined function $h\colon Y\rightarrow \int$ with $h([x])=x^m$?
\bigskip
 

 

\noindent{\bf 4} {\sl (15 points)}  Let $R$ be a commutative ring
\smallskip (a) Prove from the ring axioms  that if $a$, $b$, $c$, $d$ are
elements of $R$, then  $$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2.$$\smallskip

(b) An element $m$ of $R$ is said
to be  a sum of two squares if there exist elements $a$ and $b$ of $R$
 with $a^2+
b^2=m$. (For instance, in $\int$,  $10=3^2+1^2$ and $9=3^2+0^2$ are   sums of
two squares but $3$ is not). Use the equation in (a) to prove that if
elements  $m$ and $n$ of $R$  are both   sums of two squares, then  $mn$ is
also  a sum of two squares.\smallskip

(c) Assume that $R$ is the field of real numbers. Express the identity in (a)
in terms of the absolute values of the complex numbers $z=a+bi$, $w=c+di$
and $zw$. 


  
 \vfill\eject\bye




