Topics in Algebra

Harold M. Edwards

Chapter Chapter 2 in Essays in Constructive Mathematics, 2022, pp 47-67 from Springer

Abstract: Abstract Galois’s theorem that elements unmoved by the Galois group are rationally known is founded on the existence of a splitting field. The first three essays of this chapter explore relation between Galois’s work and Kronecker’s assertion that every field of algebraic quantities is a root field in the sense of Chapter 1 . The algorithmic description of fields of algebraic quantities in terms of adjunction relations gives an explicit construction of the splitting field of a polynomial. This is applied in the fourth essay to prove Galois’s assertion that the group of a polynomial whose coefficients are letters is the full symmetric group. These ideas stem from Kronecker, as does the fundamental theorem of divisor theory in the final essay of the chapter.

Date: 2022
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DOI: 10.1007/978-3-030-98558-5_2

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