Abstract
Algebraic and transcendental numbers are defined. Dimension and linear dependence are used to show that the set of algebraic numbers forms a sub-field of \(R\). The Steinitz exchange lemma is proven. It is shown that the minimal sub-field of \(R\) containing \(\sqrt{2}\) is smaller than the set of algebraic numbers. Liouville’s constant is discussed as an example of a real number that is not algebraic.
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© 2015 Springer International Publishing Switzerland
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Joshi, M. (2015). Linear Dependence, Fields and Transcendence. In: Proof Patterns. Springer, Cham. https://doi.org/10.1007/978-3-319-16250-8_8
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DOI: https://doi.org/10.1007/978-3-319-16250-8_8
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16249-2
Online ISBN: 978-3-319-16250-8
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