Exercise 11, § 5

Let K be a field of characteristic $\neq 2$; consider a polynomial $g(X) \in K[X]$ of even degree $\geqslant 2$, having all its roots distinct (in an algebraically closed extension of K) and such that g(0) ≠ 0. Put B = K[X], A = B[Y]/(Y² − f(X)) with f(X) = Xg(X).

a) Let y be the class of Y in A. Now prove that every a ∈ A can be written in a unique way a = P + yQ, with P and Q in B. Let $\bar{a} = P - yQ$, N(a) = a\bar{a} = P² − fQ². Now prove that the relation N(a) = 0 entails a = 0; deduce that A is an integral ring. Now prove that the invertible elements of A are the elements ≠ 0 of K.

b) Let m be the ideal AX + Ay of A; now prove that m is maximal and that the ideal $m_m$ of $A_m$ is generated by y (remark that one has $X = y^2/g(X)$). Deduce that m is an invertible A-module (use th. 1). Now prove 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 therefore not reduced to the identity element. (If one had m = At, one would have X = $\lambda t^2$ with $\lambda \in K$, whence $\lambda^{-1}X = P^2 + f/Q^2$ with P ∈ B and Q ∈ B; prove that such a relation is impossible).