Abstract
Beginning with the familiar example of the real Cartesian plane, we show how to construct a geometry satisfying Hilbert’s axioms over an abstract field. The axioms of incidence are valid over any field (Section 14). For the notion of betweenness we need an ordered field (Section 15). For the axiom (C1) on transferring a line segment to a given ray, we need a property (*) on the existence of certain square roots in the field F. To carry out Euclidean constructions, we need a slightly stronger property (**)-see Section 16.
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© 2000 Robin Hartshorne
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Hartshorne, R. (2000). Geometry over Fields. In: Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-22676-7_4
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DOI: https://doi.org/10.1007/978-0-387-22676-7_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3145-0
Online ISBN: 978-0-387-22676-7
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