Valuation Rings and Rigid Elements in Fields

Ware, Roger

Canadian journal of mathematics, Tome 33 (1981) no. 6, pp. 1338-1355

Résumé

In [20], T. A. Springer proved that if A is a complete discrete valuation ring with field of fractions F, residue class field of characteristic not 2, and uniformizing parameter π then any anisotropic quadratic form q over F has a unique decomposition as q = q 1 ⊥ 〈π〉q 2, where q 1 and q 2 represent only units of A, modulo squares in F (compare [14, Satz 12.2.2], [19, §4], [18, Theorem 8.9]). Consequently the binary quadratic form x 2 + πy 2 represents only elements in Ḟ 2 ∪ πḞ 2, where Ḟ 2 denotes the set of nonzero squares in F. Szymiczek [21] has called a nonzero element a in a field F rigid if the binary quadratic form x 2 + ay 2 represents only elements in Ḟ 2 ∪ aḞ 2.

Détail

DOI : 10.4153/CJM-1981-103-0

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