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\begin{center} {\Large\bf Algebra V63.0344 \\ Assignment 3} \\ Due
Valentine's Day, in Recitation \end{center}
\beq
What's terrible is to pretend that second-rate is first-rate.
To pretend that you don't need love when you do; or you like
your work when you know quite well you're capable of better.
\\ Doris Lessing, 1919-2013
\enq
\ben
\item Let $F\subset K$, both fields, and consider $K$ as a vector space over
$F$. Let $\ah\in K$, $\ah\neq 0$. Prove that the map $T_{\ah}:K\ra K$ given by
$T_{\ah}(\beta)=\ah\beta$ is a homomorphism. Prove further that it is
an isomorphism between $K$ and itself. (To clarify, you are to show that
$T_{\ah}$ is a {\em vector} homomorphim. For that you need show that it
sends sums to sum, differences to differences, and scalar products to
scalar products. The last means you need $T_{\ah}(\lam k) = \lam T_{\ah}(k)$
for every $\lam\in F$ and every $k\in K$.)
\item Let $F=Q[x]/(x^3-2)$. Find the inverse of $2+x+x^2$ by calling it
$a+bx+cx^2$ and getting, and solving, three equations in three unknowns.
\item Let $F=Z_7[x]/(x^3-2)$. Find the inverse of $2+x+x^2$ by calling it
$a+bx+cx^2$ and getting, and solving, three equations in three unknowns.
\item For which $n\geq 2$ and which $a\in Z$ can we use Eisenstein's
criterion to show $x^n-a$ is irreducible in $Q[x]$? (As three examples,
how about $n=6, a= 12$ and how about $n=4$ and $a=-4$ and how about
$n=11$ and $a=72$?)
\item {\bf DO NOT SUBMIT} Let $V,W$ be vector spaces over the same field $F$. Let $HOM[V,W]$
(this is standard) denote the set of homomorphisms $T:V\ra W$. Endow
$HOM[V,W]$ with a natural addition and scalar multiplication so that
it becomes a vector space. Let $\vec{v_1},\ldots,\vec{v_m}$ and
$\vec{w_1},\ldots,\vec{w_n}$ denote bases for $V,W$ respectively.
Describe a natural basis for $HOM[V,W]$.
\item (Just for Fun:)
What movie title describes an object
in $HOM[V,W]$? (Hint: The movie came out when you were an $\eps$.)
\pagebreak
\item By $C^*$ we mean $C-\{0\}$, the nonzero complex numbers.
Here we examine $(C^*,\cdot)$, the group under
multiplication. Let $S$ be the set of solutions to the equation $z^{12}=1$.
\ben
\item Draw a nice picture of $S$ on the complex plane $C$.
\item For each $z\in S$ marked above, give the {\em minimal} positive integer $s$ with $z^s=1$.
\item Prove that $S$ is a subgroup of $C^*$
\item Prove that $S$ is the {\em only} subgroup of $C^*$ with (precisely) twelve elements.
\een
\een
\beq
Mathematics, rightly viewed, possesses not only truth,
but supreme beauty - a beauty cold and austere, like
that of sculpture, without appeal to any part of our
weaker nature, without the gorgeous trappings of
painting or music, yet sublimely pure, and capable
of a stern perfection such as only the greatest art
can show. The true spirit of delight, the exaltation,
the sense of being more than Man, which is the touchstone
of the highest excellence, is to be found in mathematics
as surely as in poetry
\\ -- Bertrand Russell,
The Study of Mathematics, 1902
\enq
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