Exercise 7, § 2

Denote by A (resp. $\mathfrak{a}$) the set of rational fractions of the form $P/(X^2+ 1)^n$, where $n$ is an integer $\geqslant 0$ and $P\in \mathbf{Q}[X]$ is a polynomial of degree $\leqslant 2n$ (resp. $\leqslant 2n-1$). a) Prove that A is a subring of the field $\mathbf{Q}(X)$ and that $\mathfrak{a}$ is an ideal of the ring A. b) The A-modules $\mathfrak{a}\oplus \mathfrak{a}$ and $A\oplus A$ are isomorphic. The A-module $\mathfrak{a}$ is projective and finitely generated but is not isomorphic to A.

c) There exists a quadratic extension K of the field $\mathbf{Q}$ such that the $A_{(K)}$-modules $\mathfrak{a}_{(K)}$ and $A_{(K)}$ are isomorphic.