Math 6310 • Algebra

Let \(\mathbb{Z}[x]\) be the ring of polynomials in \(x\) with integer coefficients. Let \(R\) denote
the subring consisting of all polynomials having their coefficients of \(x\) and \(x^2\) equal to 0.

\begin{enumerate}
\item[(a)]
Show that \(\mathbb{Q}(x)\) is the field of fractions of \(R\).
\item[(b)]
Compute the integral closure of \(R\) in \(\mathbb{Q}(x)\).
\item[(c)]
Does there exist a polynomial \(g(x)\in R\) such that \(R\) is generated as a ring by 1 and \(g(x)\)?
\end{enumerate}