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\noindent{\bf Math 431 homework}\hfill Due Friday, September 7, 2001

\ex (Dummit and Foote) If $x,y,z$ are real numbers such that
$$x+y+z = 1, \quad x^2 + y^2 + z^2 = 2, \quad x^3 + y^3 + z^3 = 3$$
show that for all $n \in \Z_{\ge 0}$, the number $x^n+y^n+z^n$ is
rational.  Find $x^4 + y^4 + z^4$.  Show that $x,y,z$ are not
rational.

\ex
What is the dimension of the vector space of all homogeneous
polynomials in $x,y,z$ of total degree~$5$?  What is the
dimension of the subspace of symmetric polynomials?  (i.e.,
polynomials unchanged under all permutations of the
three variables)

\ex
Let $e_i$ denote the $i$-th elementary symmetric function of
$n$ variables $x_1, \cdots , x_n$.  (Implicitly, $e_i = 0$ for
$i &gt; n$.)  Let 
$$p_k = \sum_{i=1}^n x_i^k$$
be the $k$-th power sum of the variables.  Verify Newton's
identities: for all positive integers~$k$,
$$\sum_{j=0}^{k-1} (-1)^j p_{k-j} e_j + \quad (-1)^k k e_k = 0.$$
Prove that the $p's$ are polynomials, with rational coefficients,
in the $e's$, and vice versa.

% \ex
% Give a two or three sentence outline of the proof that
% a polynomial of degree~$n$ with coefficients in a field
% has at most $n$ roots.
% 
% \ex
% Let $G$ be a finite subgroup of the multiplicative group~$F^*$
% of a field~$F$.  Prove that $G$ is cyclic by filling in the
% details of the following argument:
% if $\#G = \prod_i p_i^{a_i}$, let $q_i := \#G/p_i$.
% There is an element $u_i$ of~$G$ that does {\bf not} satisfy
% the equation $x^{q_i} = 1$.  If $u$ and $b$ have relatively
% prime orders in an abelian group~$G$ then the order of their
% product is the product of their orders.  Therefore the
% element
% $$ u = \prod_i u_i^{\#G/p_i^{a_i}}$$
% has order $\#G$, and $G$ is cyclic as claimed.

\ex
Find, with proof, the following degrees:
\begin{itemize}
\item $[\Q(\sqrt{2}):\Q]$
\item $[\Q(\sqrt{2}+\sqrt{3}):\Q]$
\item $[\Q(\sqrt{3+4i}):\Q]$, where, here, $i = \sqrt{-1}$
\item $[\Q(\sqrt{3+2\sqrt{2}}):\Q]$
\item $[\Q(\sqrt{3+4i} +\sqrt{3-4i}:\Q]$, where, for the sake of
definiteness, we take the first square root to be the one that lies
in the first quadrant of the complex plane, and the second square
root to the the one that lies in the fourth quadrant.
\end{itemize}

\end{document}

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