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Preliminary Exam - Spring 1996
Problem 2
Let
![% latex2html id marker 670
$K \subset \mbox{$\mathbb{R}^{n}$}$](img2-48.gif)
be compact and
![$\lbrace B_j \rbrace_{j=1}^\infty$](img3-48.gif)
be a sequence of open balls which covers
![$K$](img4-48.gif)
.
Prove that there is
![$\varepsilon > 0$](img5-48.gif)
such that every
![$\varepsilon$](img6-48.gif)
-ball
centered at a point of
![$K$](img7-48.gif)
is contained in one of the balls
![$B_j$](img8-48.gif)
.
Problem 4
Let
![$r < 1 < R$](img10-48.gif)
. Show that for all sufficiently small
![$\varepsilon >0$](img11-48.gif)
,
the polynomial
has exactly five roots (counted with their multiplicities) inside the annulus
Problem 5
Prove or disprove: For any 2
![$\times $](img14-48.gif)
2 matrix
![$A$](img15-48.gif)
over
![% latex2html id marker 722
$\mbox{$\mathbb{C}\,^{}$}$](img16-48.gif)
, there
is a 2
![$\times $](img17-48.gif)
2 matrix
![$B$](img18-48.gif)
such that
![$A=B^2$](img19-48.gif)
.
Problem 6
If a finite homogeneous system of linear equations with rational coefficients
has a nonzero complex solution, need it have a nonzero rational solution?
Prove or give a counterexample.
Problem 7
Prove that
![$f(x)=x^4+x^3+x^2+6x+1$](img20-48.gif)
is irreducible over
![% latex2html id marker 748
$\mbox{$\mathbb{Q}\,^{}$}$](img21-48.gif)
.
Problem 8
Determine the rightmost decimal digit of
Problem 9
Exhibit infinitely many pairwise nonisomorphic quadratic extensions of
![% latex2html id marker 770
$\mbox{$\mathbb{Q}\,^{}$}$](img23-48.gif)
and show they are pairwise nonisomorphic.
Problem 10
Show that a positive constant
![$t$](img24-48.gif)
can satisfy
if and only if
![$t<e$](img26-48.gif)
.
Problem 11
Suppose
![$\varphi$](img27-48.gif)
is a
![$C^1$](img28-48.gif)
function on
![% latex2html id marker 800
$\mbox{$\mathbb{R}^{}$}$](img29-48.gif)
such that
Prove or give a counterexample:
![$b$](img31-48.gif)
must be zero.
Problem 12
Let
![$M_{2 \times 2}$](img32-48.gif)
be the space of 2
![$\times $](img33-48.gif)
2 matrices over
![% latex2html id marker 835
$\mbox{$\mathbb{R}^{}$}$](img34-48.gif)
,
identified in the usual way with
![% latex2html id marker 839
$\mbox{$\mathbb{R}^{4}$}$](img35-48.gif)
.
Let the function
![$F$](img36-48.gif)
from
![$M_{2 \times 2}$](img37-48.gif)
into
![$M_{2 \times 2}$](img38-48.gif)
be defined by
Prove that the range of
![$F$](img40-48.gif)
contains a neighborhood of the origin.
Problem 13
Let
![$f=u+iv$](img41-48.gif)
be analytic in
a connected open set
![$D$](img42-48.gif)
, where
![$u$](img43-48.gif)
and
![$v$](img44-48.gif)
are real valued.
Suppose there are real constants
![$a$](img45-48.gif)
,
![$b$](img46-48.gif)
and
![$c$](img47-48.gif)
such that
![$a^2+b^2 \not= 0$](img48-48.gif)
and
in
![$D$](img50-48.gif)
. Show that
![$f$](img51-48.gif)
is constant in
![$D$](img52-48.gif)
.
Problem 14
Suppose
![% latex2html id marker 894
$f\colon [0,1]\to \mbox{$\mathbb{C}\,^{}$}$](img53-48.gif)
is continuous. Show that
defines a function
![$g$](img55-48.gif)
that is analytic everywhere in the complex plane.
Problem 15
Suppose that
![$A$](img56-48.gif)
and
![$B$](img57-48.gif)
are real matrices such that
![$A^t=A$](img58-48.gif)
,
for all
![% latex2html id marker 920
$v \in \mbox{$\mathbb{R}^{n}$}$](img60-48.gif)
and
Show that
![$AB=BA=0$](img62-48.gif)
and give an example where neither
![$A$](img63-48.gif)
nor
![$B$](img64-48.gif)
is zero.
Problem 16
Let
![$A$](img65-48.gif)
be the
![$n \times n$](img66-48.gif)
matrix which has zeros on the main diagonal
and ones everywhere else. Find the eigenvalues and eigenspaces of
![$A$](img67-48.gif)
and
compute
![$\det(A)$](img68-48.gif)
.
Problem 17
Let
![$G$](img69-48.gif)
be the group of 2
![$\times $](img70-48.gif)
2 matrices with determinant
![$1$](img71-48.gif)
over the four-element field
![% latex2html id marker 960
$\mbox{\bf {F}}$](img72-48.gif)
.
Let
![$S$](img73-48.gif)
be the set of lines through the origin in
![% latex2html id marker 964
$\mbox{\bf {F}}^2$](img74-47.gif)
.
Show that
![$G$](img75-47.gif)
acts
faithfully on
![$S$](img76-47.gif)
.
(The action is faithful if the only element of
![$G$](img77-47.gif)
which fixes
every element of
![$S$](img78-47.gif)
is the identity.) _group action, faithful
Problem 18
Let
![$G$](img79-47.gif)
and
![$H$](img80-47.gif)
be finite groups of relatively prime orders.
Show that the automorphism group
![$\mathrm{Aut}(G\times H)$](img81-46.gif)
is isomorphic
to the direct product of
![$\mathrm{Aut}(G)$](img82-46.gif)
and
![$\mathrm{Aut}(H)$](img83-45.gif)
.
Previous: Fall95
Next: Fall96
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10