Sums of Hermitian squares
The level of a commutative ring R is the least integer n for which \(-1\) is a sum of n squares in R. In this paper the author defines the hermitian level of a ring with involution \(x\to \bar x\) in the same way except that he considers hermitian squares \(x\bar x\) instead of squares. Among others the following results are proved. There exist commutative rings with non-trivial involution having any positive integer as hermitian level. For fields with non-trivial involution or quaternion division algebras with the standard involution, the level, if finite, is a power of two. The case of a quaternion division algebra with a non-standard involution is left open. The paper ends with some remarks about the finiteness of the level for quaternion division algebras over ordered fields.
scientific article; zbMATH DE number 4049716
scientific article