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\begin{center} {\Large\bf Algebra V63.0349 \\ Assignment 3} \\ Due
February 12, 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 ({\bf Note:} We will cover this next week. Not to be submitted.)
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. That is, show
\ben
\item $T_{\ah}(v_1+v_2)=T_{\ah}(v_1)+T_{\ah}(v_2)$ for all $v_1,v_2\in K$
\item $T_{\ah}(\lam v) = \lam T_{\ah}(v)$ for all $v\in K$, $\lam\in F$.
\item
Prove further that $T_{\ah}$
an isomorphism between $K$ and itself.
\een
\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 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
\item Set $Z[\sqrt{-2}]=\{a+b\sqrt{-2}\}$. This is a Euclidean Domain (a nice exercise but
not requested) with $d(\ah) = |\ah|^2=a^2+2b^2$. Assumming that show:
\ben
\item If $d(\ah)$ is an integer prime then $\ah$ is a prime.
\item If $p$ is an integer prime and there $p=x^2+2y^2$ has an integer solution then $p$
is {\em not} a prime in $Z[\sqrt{-2}]$.
\item If $p$ is an integer prime and there $p=x^2+2y^2$ has no integer solution then $p$
is a prime in $Z[\sqrt{-2}]$.
\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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