Valuation Rings and Rigid Elements in Fields

Abstract

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 = q1 ⊥ 〈π〉q2, where q1 and q2 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 x2 + πy2 represents only elements in Ḟ2 ∪ πḞ2, where Ḟ2 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 x2 + ay2 represents only elements in Ḟ2 ∪ aḞ2.