uniformizing parameter
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1981 ◽  
Vol 33 (6) ◽  
pp. 1338-1355 ◽  
Author(s):  
Roger Ware

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.


1978 ◽  
Vol 30 (03) ◽  
pp. 512-517 ◽  
Author(s):  
Klaus G. Fischer

Throughout this paper, it is assumed that A is the complete, equicharacteristic, local ring of an algebraic curve at a one-branch singularity whose residue field is algebraically closed and contained in A. Hence, the domain A is dominated by only one valuation ring in its quotient held F, and if t is a uniformizing parameter, then the integral closure of A in F, denoted by Ā, is [[t]].


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