Abstract
This chapter will discuss the classification problem for inner product spaces over the ring Z of rational integers. All inner products are to be symmetric. Our presentation is based on the classical theorem of Minkowski concerning lattice points in a convex symmetric subset of R n. This theorem is first used to classify inner product spaces of rank ≦4 over Z. Making use of the Hasse-Minkowski theorem (which we do not prove), it is shown that an indefinite inner product space over Z is completely determined by its rank, type, and signature; where the type is defined to be either I or II according as the space does or does not contain a vector of odd norm. It follows that the Witt ring W(Z) is isomorphic to Z.
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© 1973 Springer-Verlag Berlin Heidelberg
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Milnor, J., Husemoller, D. (1973). Symmetric Inner Product Spaces over Z. In: Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 73. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88330-9_2
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DOI: https://doi.org/10.1007/978-3-642-88330-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-88332-3
Online ISBN: 978-3-642-88330-9
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