Abstract
This chapter illustrates the general theory of Galois extensions in a special case. We study field extensions, mostly of the rational numbers, generated by the roots of 1. Even if such fields are simple to describe in purely algebraic terms, they are rich as mathematical objects. We explore some of their properties, which find different applications in number theory and algebra. Among many applications, there is a proof of a special case of Dirichlet’s theorem on primes in arithmetic progression using the cyclotomic polynomials. This chapter also includes an exercise on a solution of the inverse Galois problem for all abelian groups.
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- 1.
Heinrich Martin Weber, 5 May 1842–17 May 1913.
- 2.
Johann Peter Gustav Lejeune Dirichlet, 13 February 1805–5 May 1859.
References
D. Gay, W.Y. Vélez, On the degree of the splitting field of an irreducible binomial, Pacific J. Math. 78 (1978), 117–120.
H. Koch, Number Theory, Algebraic Numbers and Functions, Graduate Studies in Mathematics vol. 24, AMS, 2000.
J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics vol. 7, Springer, 2006.
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Brzeziński, J. (2018). Cyclotomic Extensions. In: Galois Theory Through Exercises. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-72326-6_10
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DOI: https://doi.org/10.1007/978-3-319-72326-6_10
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Online ISBN: 978-3-319-72326-6
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