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Quaternions as reflexive skew fields. (English) Zbl 1222.12003
From the text: Throughout this text a skew field, or a sfield, is a ring with a unit element 1, in which every non zero element has an inverse. Given \(D\) an sfield with centre \(F\) and \(\sigma\) an involution of \(D\), i.e. a transformation \(x\mapsto \sigma(x)\) of \(D\) for which \(\sigma(x + y) = \sigma(x) + \sigma(y)\), \(\sigma(xy) = \sigma(y)\sigma(x)\) and \(\sigma^2(x) = x\), for any \(x, y\in D\).
Jean Dieudonné, in his work on the structure of unitary groups, says in [Sur les groupes classiques. (Paris: Hermann) (1948; Zbl 0037.01304), p. 72] that \(D\) is a reflexive sfield if all the elements of \(F\) are fixed by \(\sigma\) and if \(x+\sigma(x)\) and \(x\sigma(x)\) belong to \(F\), for any \(x\in D\). He shows (by completing his proof in [Am. J. Math. 75, 665–678 (1953; Zbl 0051.01803)]) that non-commutative reflexive sfields are generalized quaternion sfields. His proof, amongst other things, relies on theorems concerning the structure of sfields of finite rank over their center, proved earlier this century in the 1920s. Now, Eliakim Hastings Moore had already stated and provided an elementary proof for the above-mentioned result provided by Dieudonné for sfields of characteristic other than 2. In 1922, E. H. Moore has defined a determinant for hermitian matrices with coefficients in those fields whose elements, fixed by the involution, are central [Bull. Am. Math. Soc. 28, 161–162 (1922; JFM 48.0128.07)].
Moore’s proof [General analysis. With the cooperation of Raymond Walter Barnard. Pt. I. Mem. Am. Philos. Soc. Vol. 1. Philadelphia: Am. Philos. Soc. (1935; Zbl 0013.11605 and JFM 61.0433.06), pp. 104–107] is based on an identity valid for any sfield with involution whose elements fixed by the involution are central. This identity, which was expressed by W. R. Hamilton himself [Elements of quaternions. (1866), p. 317; reprinted Cambridge (2009; Zbl 1204.01046)] for usual quaternions, is, in this case a translation of the identity \[ \langle\vec z\times\vec t, \vec y\rangle \vec x -\langle\vec z\times\vec t, \vec x\rangle \vec y - \langle\vec x\times\vec y, \vec t\rangle \vec z +\langle\vec x\times\vec y, \vec z\rangle \vec t = 0 \tag{1} \] in which \(x,y,z\) and \(t\) are vectors in usual three-dimensional space, and \(\langle\vec x, \vec y\rangle\) and \(\langle\vec x\times\vec y\rangle\) are, respectively, the scalar and vector products of \(x\) and \(y\). Here we sketch out Dieudonné’s non elementary proof, present Moore’s proof because the book cited above is difficult to obtain, and also present another elementary proof based on the fact that if \(V = \{x\in D\mid \sigma(x) = -x\}\), then \(V\) is a vector space on the subfield \(F_0\) of the elements of \(F\) fixed by \(\sigma\) and the application \(x\mapsto x^2\) is a regular quadratic form on that vector space.
MSC:
12E15 Skew fields, division rings
16K20 Finite-dimensional division rings
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