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The invariant class of a binary form. (La classe invariante d’une forme binaire.) (French) Zbl 1038.11073
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 [D. Simon, Indag. Math., New Ser. 12, 505–517 (2001; Zbl 1020.11065)]. The classical reference for \(n= 3\) is B. N. Delone and D. K. Faddeev [Theory of irrationalities of the third degree (Russian), Tr. Mat. Inst. Steklov. 11 (1940; JFM 61.0090.01)].

MSC:
11R29 Class numbers, class groups, discriminants
11E76 Forms of degree higher than two
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References:
[1] Cohen, H., A course in computational algebraic number theory, Graduate texts in math., 138, (1996), Springer-Verlag, Third corrected printing
[2] Cox, D.A., Primes of the form x2+ny2, (1989), Wiley New York
[3] J.E. Cremona, Classical invariants and 2-descent on elliptic curves, J. Symbolic Comput. 31 (1-2) 71-87 · Zbl 0965.11025
[4] Delone, B.N.; Faddeev, D.K., Theory of irrationalities of third degree, Trudy mat. inst. Steklov, 11, (1940) · Zbl 0061.09001
[5] ftp://megrez.math.u-bordeaux.fr/pub/numberfields/
[6] Simon, D., The index of nonmonic polynomials, Indag. math. (N.S), 12, 4, 505-517, (2001) · Zbl 1020.11065
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