Exercise 11, § 5

Let $K$ be a field of characteristic $\neq 2$; consider a polynomial $g(X) \in K[X]$ of even degree $\geqslant 2$ with all its roots distinct (in an algebraically closed extension of $K$) and satisfying $g(0) \neq 0$. Set $B = K[X]$,

where $f(X) = Xg(X)$.

(a) Let $y$ be the class of $Y$ in $A$. Show that every $a \in A$ may be written uniquely in the form $a = P + yQ$, where $P$ and $Q$ are in $B$. Let $\bar{a} = P - yQ$ and $N(a) = a \bar{a} = P^2 - f Q^2$. Show that the relation $N(a) = 0$ implies $a = 0$; deduce that $A$ is an integral domain. Show that the invertible elements of $A$ are the elements $\neq 0$ of $K$.

(b) Let $m$ be the ideal $AX + Ay$ of $A$; show that $m$ is maximal and that the ideal $m_m$ of $A_{m}$ is generated by $y$ (note that $X = y^2 / g(X)$). Deduce that $m$ is an invertible $A$-module (use Theorem 1). Show that $m^2 = AX$; deduce that $m$ is not a principal ideal of $A$ and that the group of classes of invertible $A$-modules in the field of fractions of $A$ is not reduced to the identity element. (If $m = At$, then $X = \lambda t^2$ where $\lambda \in K$, whence $\lambda^{-1}X = P^2 + f Q^2$ where $P \in B$ and $Q \in B$; prove that such a relation is impossible.)