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\begin{center} {\Large\bf Algebra V63.0349 \\ Assignment 4} \\ Due,
Friday, Februray 27 in Recitation \end{center}
\beq
The universe is not only queerer than we suppose but queerer than we {\em can}
suppose. \\ -- J.B.S. Haldane
\enq
\ben
\item Let $\ah\in C$ be a root of $x^3+x+3$. (This cubic has no special properties.)
Write $\ah^i$ in the form $a+b\ah+c\ah^2$, $a,b,c\in Q$, for $3\leq i\leq 6$.
Set $\beta=\ah^2$. Find a cubic in $Q[x]$ that has $\beta$ as a root.
\item Let $p(x)\in Z[x]$ be a {\em monic} polynomial of degree $n$, irreducible over $Q$.
Write $p(x)=x^n+b_{n-1}x^{n-1}+\ldots+b_0$.
Let $\ah\in C$ be a root of the equation $p(x)=0$. Set
\[ R= \{a_0+a_1\ah+\ldots+a_{n-1}\ah^{n-1}: a_0,\ldots,a_{n-1}\in Z\} \]
Show that $R$ is a ring. (The hard part is closure under multiplication.) Now
suppose further that $p(x)$ has constant term $b_0=\pm 1$. Show that $\ah^{-1}\in R$.
\item Here we examine the polynomial $p(x)=x^4+1$. Let $\ah,\beta,\gam,\del$ denote
the complex roots of $p(x)=0$.
\ben
\item Find $\ah,\beta,\gam,\del$ both in terms of polar coordinates $\ah=re^{i\theta},\ldots$
(this is actually easier for this particular problem) and in the Cartesian $\ah=a+bi,\ldots$ forms
and mark them on the complex plane.
\item Give the factorization of $p(x)$ into irreducibles in $C[x]$.
\item Give the factorization of $p(x)$ into irreducibles in $Re[x]$.
\item Give the factorization of $p(x)$ into irreducibles in $(Q(\sqrt{2}))[x]$.
\item Give the factorization of $p(x)$ into irreducibles in $(Q(i\sqrt{2}))[x]$.
\item Show $p(x)$ is irreducible in $Q[x]$ {\em using} the following idea: If, say, $p(x)=f(x)g(x)$,
then, as $p(x)$ factors into four linear factors in the first part above, $f(x)$ and $g(x)$ must be
a product of some (but not all) of those factors. Try all possiblities for products of the linear factors
(there aren't that many) and check that none of them give an $f(x)\in Q[x]$.
\item There are many ways to show that a polynomial is irreducible over $Q[x]$. Show $p(x)$ is
irreducible over $Q[x]$ by some other method -- your choice!
\een
\item Let $f(z)=10+(i+1)z^4+6z^5$. (Here $i=\sqrt{-1}$.) Find an angle $\phi$ (following the
proof of the Fundamental Theorem of Algebra) such that
$|f(10^{-100}e^{i\phi})| < 10 - 1.4\cdot 10^{-400}$.
\item Let $\ah\in C$ satisfy $\ah^3+\beta\ah^2+\gam\ah+\del=0$ where $\beta,\gam,\del$
satisfy cubics with coefficients in $Q$. (That is, $\beta^3+a\beta^2+b\beta+c=0$ for some
$a,b,c\in Q$, and similarly for $\gam,\del$.)
(Possibly smaller degree
polynomials are satisfied by these numbers.)
\ben
\item Give an upper bound for the degree of
$\ah$ over $Q$.
\item Show that the minimal polynomial of $\ah$ over $Q$ does {\em not} have degree
precisely five.
\een
\item If $\ah,\beta\in K$ are algebraic over $F$ of degrees $m$ and $n$ respectively
and if $m,n$ are relatively prime, prove that $F(\ah,\beta)$ is of degree $mn$ over
$F$.
\item Let $p$ be an odd prime. Assume as a (true!) fact that when $\ah,\beta\in Z_p^*$
are {\em not} squares then $\ah/\beta$ is a square.
Let $L,K$ be fields, both of size precisely $p^2$.
Using the results on extension of degree two show that $L,K$ are isomorphic.
($L,K$ are isomorphic if there exists a bijection $\psi: L\ra K$
that preserves addition and multiplication.)
\een
\beq
Nothing is more fruitful - all mathematicians know it - than those
obscure analogies, those disturbing reflections of one theory in
another; those furtive caresses, those inexplicable discords; nothing
also gives more pleasure to the researcher. The day comes when the
illusion dissolves; the yoked theories reveal their common source
before disappearing. As the {\em Gita} teaches, one achieves
knowledge and indifference at the same time.
\\ Andr\'e Weil
\\ (Note: ``indifference" is a controversial translation of the original Sanskrit, ``detachment" is often
used instead)
\enq
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