Abstract
This chapter describes a kind of field extension most common in Galois theory: algebraic extensions, i.e. extensions K ⊆ L in which each element α ∈ L is a zero of a nontrivial polynomial with coefficients in K. We relate the elements of algebraic extensions to the corresponding polynomials and look at the structures of the simplest extensions K(α) of K. We also introduce the notion of the degree of a field extension and prove some of its properties such as the Tower Law.
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Notes
- 1.
Jacob Lüroth, 18 February 1844–14 September 1910.
- 2.
Trygve Nagell (Nagel), 13 July 1895–24 January 1988.
- 3.
For a proof of transcendence of these numbers see [L], Appendix 1.
References
P.M. Cohn, Algebra, vol. 3, Second Edition, John Wiley and Sons, 1991.
N. Jacobson, Basic Algebra II, Second Edition, Dover Books on Mathematics (W.H. Freeman and Company), 1989.
S. Lang, Algebra, Third Edition, Addison-Wesley, 1993.
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Brzeziński, J. (2018). Algebraic Extensions. In: Galois Theory Through Exercises. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-72326-6_4
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DOI: https://doi.org/10.1007/978-3-319-72326-6_4
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Online ISBN: 978-3-319-72326-6
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