The invariant class of a binary form (Q1871528)

The author considers how irreducible binary forms of any degree \(n\) over a ring \(R\) could be made to correspond to an ideal class in the field \(K\) of the root, so as to be invariant under \(\text{SL}(2,\mathbb{R})\). In analogy with the quadratics, a module of basis elements in \(K\) corresponds uniquely to an ideal class (but neither biuniquely when \(n> 3\) nor surjectively when \(n> 2\) , as shown by examples). The author's earlier work is cited here [\textit{D. Simon}, Indag. Math., New Ser. 12, 505--517 (2001; Zbl 1020.11065)]. The classical reference for \(n= 3\) is \textit{B. N. Delone} and \textit{D. K. Faddeev} [Theory of irrationalities of the third degree (Russian), Tr. Mat. Inst. Steklov. 11 (1940; JFM 61.0090.01)].