Separable free quadratic algebras over quadratic integers
The authors determine the isomorphim classes of free separable quadratic algebras over the ring of integers \(R\) of a real quadratic number field \(K\) in terms of properties of the fundamental unit of \(R\). According to [\textit{A. J. Hahn}, Quadratic algebras, Clifford algebras, and arithmetic Witt groups, Springer, New-York (1994; Zbl 0791.11021)], a free quadratic algebra \(S= R[X]/(X^2-pX-q)\) over a commutative ring \(R\) is separable (or étale) if and only its discriminant \(\Delta_S = p^2+4q\) is inversible in \(R\); and two free such algebras \(S_1\), \(S_2\) are isomorphic if and only if the quotient \(\Delta_{S_1}/\Delta_{S_2}\) is a square in \(R^\times\). Moreover in case \(R\) is an integrally closed domain of characteristic \(\operatorname{char} \neq 2\), the canonical morphism induced by the discriminant on the group of isomorphim classes of free quadratic separable algebras \(\delta_R : Q_f(R) \rightarrow R^\times/R^{\times 2}\) is injective. So, in case \(R\) is the ring of integers of a real quadratic number field \(K\), the group \(Q_f(R)\) is the trivial group 1, the cyclic group \(C_2 = \mathbb Z / 2\mathbb Z\), or the Vierergruppe \(V_4 = \mathbb Z / 2\mathbb Z \times \mathbb Z / 2\mathbb Z\). In the paper under review, the authors correct earlier results of \textit{J.-D. Thérond} [C. R. Acad. Sci., Paris, Sér. A 281, 939--942 (1975; Zbl 0343.13003)] and give a complete characterization of \(Q_f(R)\) which involves both the discriminant \(d_K\) of the real quadratic field \(K\) and the fundamental unit \(\varepsilon\) of \(R\). As a consequence of their results, they point out that \(K\) has an unramified quadratic extension \(L\) with ring of integers free over \(R\) if and only if \(\varepsilon\) has norm +1.
scientific article; zbMATH DE number 2136817
scientific article