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A History of Abstract Algebra pp 263–273Cite as

Invariant Theory and Polynomial Rings

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Part of the book series: Springer Undergraduate Mathematics Series ((SUMS))

Abstract

This chapter is concerned with another, related branch of algebra that mingles polynomials with geometry: invariant theory. We shall see how this field was decisively rewritten by the young David Hilbert in work that made his name in the international mathematical community.

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Notes

  1. 1.

    The competition turned into a famous scandal. The English mathematician Henry Smith wrote to the organisers to point out that he had already published a complete solution to that precise question some years before. Next it was suggested, entirely falsely, that Minkowski had entered the competition corruptly, knowing of Smith’s work. Then Smith died. The only way out for the French was to proclaim Minkowski and Smith joint winners, which they duly did.

  2. 2.

    See (Blumenthal 1935, p. 390).

  3. 3.

    This account follows (McLarty 2012).

  4. 4.

    This claim is odd: there is no terminating algorithm for finding the lowest power of x, although it is clearly ≤ r 1.

  5. 5.

    That is, homogeneous polynomials.

References

  • Blumenthal, O.: Lebensgeschichte. In: Hilbert, D. (ed.) Gesammelte Abhandlungen, vol. 3, pp. 388–429. Springer, Berlin (1935)

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  • Frei, G. (ed.): Der Briefwechsel David Hilbert – Felix Klein (1886–1918). Vandenhoeck & Ruprecht, Göttingen (1985)

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  • Hilbert, D.: Ueber die Theorie der algebraischen Formen. Math. Ann. 36, 473–534 (1890); in Gesammelte Abhandlungen 2, 199–257

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  • Hilbert, D.: Ueber die vollen Invariantensysteme. Math. Ann. 42, 313–373 (1893); in Gesammelte Abhandlungen 2, 287–344

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  • McLarty, C.: Theology and its discontents: David Hilbert’s foundation myth for modern mathematics. In: Doxiades, A., Mazur, B. (eds.) Circles Disturbed: The Interplay of Mathematics and Narrative, pp. 105–129. Princeton University Press, Princeton (2012)

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Authors and Affiliations

  1. School of Mathematics and Statistics, The Open University, Milton Keynes, UK

    Jeremy Gray

  2. Mathematics Institute, University of Warwick, Coventry, UK

    Jeremy Gray

Authors
  1. Jeremy Gray
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Gray, J. (2018). Invariant Theory and Polynomial Rings. In: A History of Abstract Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-94773-0_25

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