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2018 | OriginalPaper | Chapter

15. What Is ‘Galois Theory’?

Author : Jeremy Gray

Published in: A History of Abstract Algebra

Publisher: Springer International Publishing

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Abstract

In this chapter, we are still concerned with the question of how Galois Theory became established as we look at Klein’s influence in more detail.

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previous chapter The Galois Theory of Hermite, Jordan and Klein

next chapter Algebraic Number Theory: Cyclotomy

It is quoted in Wussing (1984, p. 274).

They are quoting from Biermann (1973/1988).

Quoted in Parshall and Rowe (1994, 201).

Recall that the metacyclic case is the one in which all the roots are rational functions of any two of them; the best case is binomial equations.

They are the elements in H ∪ (13)H.

In modern terms, a natural irrational lies in the splitting field of the original polynomial.

Petersen was a very energetic producer of textbooks for students in Copenhagen.

See Chap. 23 below, where it is explained that these are roots of polynomial equations with coefficients taken modulo a prime; in modern terms algebraic extensions of a finite field.

An equation was said to be normal if each root of it is expressible as a polynomial function with integer coefficients of any one of them.

See the paper Hulpke (1999) and the pdf of a talk by Hulpke available at

http://www.math.colostate.edu/~hulpke/talks/galoistalk.pdf.

See his ‘Galois groups as permutation groups’ and his ‘Recognizing Galois group s S _n and A _n’, and other good sources on the web. Dedekind’s original proof has been tightened in many places; one due to Tate is on the web. http://www.math.mcgill.ca/labute/courses/371.98/tate.pdf.

Biermann, K.R.: Die Mathematik und ihre Dozenten an der Berliner Universität, 1810-1920. Stationen auf dem Wege eines mathematischen Zentrums von Weltgeltung. Akademie-Verlag, Berlin (1973/1988)

Borel, É., Drach, J.: Introduction à l’étude de la théorie des nombres et à l’algèbre supérieure. Librarie Nony, Paris (1895)

Cox, D.A.: Galois Theory, 2nd edn. 2012. Wiley Interscience, New York (2004)

Hölder, O.: Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen. Math. Ann. 34, 26–56 (1889)

Hölder, O.: Galois’sche Theorie mit Anwendungen. Encyklopädie der mathematischen Wissenschaften 1, 480–520 (1899)

Hulpke, A.: Galois groups through invariant relations. Groups St. Andrews 1997 Bath, II. London Math. Soc. Lecture Note Series, vol. 261, pp. 379–393. Cambridge University Press, Cambridge (1999)

Netto, E.: Substitutionentheorie und ihre Anwendungen auf die Algebra. Teubner, Leipzig (1882); English transl. The Theory of Substitutions and its Applications to Algebra. P.N. Cole (transl.) The Register Publishing Company, 1892

Parshall, K., Rowe, D.E.: The Emergence of the American Mathematical Research Community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore. American Mathematical Society/London Mathematical Society, Providence (1994)

Petri, B., Schappacher, N.: From Abel to Kronecker: Episodes from 19th Century Algebra. In: Laudal, O.A., Piene, R. (eds.) The legacy of Niels Henrik Abel: The Abel bicentennial 2002, pp. 261–262. Springer, Berlin (2002)

Wiman, A.: Endliche Gruppen linearen Substitutionen. Encyklopädie der mathematischen Wissenschaften 1, 522–544 (1900)

Wussing, H.: The Genesis of the Abstract Group Concept, transl. A. Shenitzer. MIT Press, Cambridge (1984)

Title: What Is ‘Galois Theory’?
Author: Jeremy Gray
Publisher: Springer International Publishing
Book: A History of Abstract Algebra
Print ISBN: 978-3-319-94772-3

Electronic ISBN: 978-3-319-94773-0

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-94773-0_15

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