Mathematics366
Midterm t (100points)
Points(leaveblank)
2
I
3
4
5
x
tr
)il;-, CfrPtu
Youcanwork on the problemsin anyorderyou like. showyour
work! All problemscarry the same
weight.
Calculationstepsandexplanationfor statements
madearea crucialpart of a solution.partial credit
will be givensparingly- rathercompleteoneproblemthan start two
onlypartially.
t)
Find an integersolutionx for the equation
23x mod89 = 4
g.r( ( Lnr s") !
3
87 = 3'23 + tO a 2'o = ffi " 2 3
3 ' 23 - 7o
e3 * | r c + 3
2' 70- 65
&0 c . 68 + <
3- Z
3 = - l 2+ ^ [ ) l=
= )l:3-?s
3-"Or-6s
.= + - ( e< - z o) - z o
= - 8 G< ) * g 3 t ? 3
axs?-sl=
=?
=) ?j-t=zl (vtel
llk = 3y Cu.<sll
23 - s ( s I
-a.za/
2) Determine(with explanation)the units and zerodivisorsin the ringzn.
h
q*(to1) * X*tLto,3J:S*,/Crpt*l *fr"*(({o,tl* I
kt
[], = ,* ,ti a ' , 'c u, '&
t =3"s
z s = q ' < e6
&r*,st|
/L,.*(
*Lq*"4
zQ C**a ro)
$J
q"< bte oLuw*,
e,*-*L<
haw ,lr,r*r*,l
&,n *Je)
%-{
\ l4^"c{,€ .t q
q*."e
3) Constructthe multiplicationtablefor the field with 4 elementsby usingpolynomialarithmetic
modulo2andmodulox2+ x +I.
In thisfield,calcutarc
ft,where
cl
\r+{
p I
t
T"
6t*"^11' 1'*e*7te v&Q*44 =frt, 4 v*/ \
Y
c0 o
t, s t
Y
a is theelement
represented
bl thepolynomial
x.
XZ= *xt-l t*,gc,tV\x*/J
I
E x.,-I U*1 27
x
hli
X (rdl = tLr. = -((x2* uttl
3 d (z)
v
cola."t&
q& *W
wll
=,) 9' *.
4*r
3
q*{
V
-f
Xrt
r*.k
lb
\*l
*
; a+l =-Kvl
x- (rc.{)=-l =r*,
* *
x,x=_t
F"",Wif*l
Wt
4\ LetR = Q[r] andletA= {/. R | /(1) = 0}.
a) Showthat A is a subringof R.
b) Showthat A is an idealin R.
c) Letg = {f e n | /(t) = l}. ShowthatB is not a subringofR.
ql Wffiq cr.-k\et '?
o*o:.;
'.rro^'
kql tr)=d{,}+q1(lJ:
t)/q*4
6 *
'){q
,# *'{ e'4
f{,f,r *:{{r ) '-frs-d)
I t,
'
S'";'{*
\
nrt # q ;C rl *fdrJtfr )= Oo=-o
[l
{U) AE t_.
tl
,
U 4 e A , t e& ^ fl "\[+S J (r)* dC t lq { r }
*!froot,
U e-'ir
: C fl trlv'
o)
odq s A
UA/ r{d*x flq dele l3
n''{
rL, (q *1A) qAo
A er\
* {U+ {[r] =-l* l; c.# t
T1'.eM & (& ta-*nt*A*p* bl VI a
l,
tr
r{l
lizfvl
lt
\,-
Y
t-
r
l- tl
\
t lt
I
+-Dt
t
5) Let R be a commutative
ring
with one,in which R and
consider
the onry ideals.For 0 .r e
theidear
qenerated
{0}
R
l
z{ = R' conclude tf,at r mustry"lrr,i.r,i. 1 =r:iu: iiTr; -.r. il;#Jn-Tio
have*-tn*rru and ti"refore
";-;..that R is a fierd"l {0}andthus
conclude
consistsof
full
;Hit"should
ti o +Ked-
"i,.*.r.
(It is not *m.*",
a-J ,4j
4 ={x%
+("" ( h %,q/b^) A*&
IL" t{t* 4rr|{
td*/e
Ld4 &Jd,
\J
rL A*f.,|,
,A- 4ee 4 - <K7n
{
Q,jf,*. ,\+ -(*
t^e tL s * "
4e= x-r
U-
to simprypoinr to Iecture
or home_
t4 q. .
IL, (ec,*-**-+.){
li q ,*-- r&
o"^-rl..bo q^% 7
t
u'r.L-"(
q. L"*o{$
-{A
"'I
rf
q cr"4'4#,
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s