Math 6310 • Algebra

Let \(n\) a positive integer, and let \(\mathbf{x}:=x_1,\ldots,x_n\) be indeterminates over
\(\mathbb{Q}\). For each positive integer \(i\) set
\[
\alpha_i=x_1^i+\cdots+x_n^i.
\]
\begin{enumerate}
\item Clearly \(\mathbb{Q}(\alpha_1,\ldots,\alpha_n)\subseteq
\mathbb{Q}(s_1,\ldots,s_n)\), the field of symmetric functions in the \(\mathbf{x}\). What is
the degree of this extension?
\item Prove the characteristic polynomial of an \(n\times n\) matrix \(A\) over \(\mathbb{Q}\) is
completely determined by the elements \(\operatorname{trace}(A),\ldots,\operatorname{trace}(A^n)\).
\end{enumerate}