Sums of hermitian squares
Sums of hermitian squares
复制标题
厄米平方和
DOI：
10.1016/0021-8693(88)90273-6
复制
发表时间：
1988
影响因子：
0.9
通讯作者：
D. W. Lewis
中科院分区：
文献类型：
--
作者：
D. W. Lewis
具有恒等的可交换环R的阶数是使-1可以表示为R中n的平方和的最小整数n，如果-1不是平方和，则R的阶数是无穷。Pfister [lo]的一个著名结果是，对于场R，如果它是有限的，能级总是2的幂。后来Dai， Lam和Peng[3]给出了以任意规定的正整数为水平的交换环的例子。本文研究具有非平凡对合的环。我们定义对合环的厄米能级的方法与能级相同，只不过我们用厄米平方代替了平方。证明了具有非平凡对合的交换环在厄米能级上具有任意正整数的存在。我们证明了对于非平凡对合域和具有标准对合的四元数除法代数，厄米能级在有限情况下是2的幂。然后研究了具有非标准对合的四元数除法代数的厄米能级。我们证明，在这种情况下，任何2的幂都可以出现，但是否可以出现除2的幂之外的整数，这是一个开放的问题。在本文的最后一节，我们得到了一个具有标准对合和非标准对合的四元数代数，其基域为有序域的厄米特能级的有限性的结果。
The level of a commutative ring R with identity is the least integer n for which-1 is expresssible as a sum of n squares in R. If-1 is not a sum of squares then the level of R is infinity. A famous result of Pfister [lo] is that for a field R the level is always a power of two if it is finite. Later Dai, Lam, and Peng [3] produced examples of commutative rings having any prescribed positive integer as level.In this paper we consider rings with non-trivial involution. We define the hermitian level of a ring with involution in the same way as the level except that we use hermitian squares instead of squares. We show that there exist commutative rings with non-trivial involution having any positive integer as hermitian level. We show that the hermitian level, if finite, is a power of two for fields with non-trivial involution and for quaternion division algebras with the standard involution. We then investigate the hermitian level for quaternion division algebras with non-standard involution. We show that any power of two can occur in this case but leave it as an open question whether integers other than powers of two can occur. In the final section of the paper we obtain a result about the finiteness of the hermitian levels for a quaternion algebra with both standard and non-standard involutions, the base field being an ordered field.