Algebraic and Transcendental Numbers

  • Antonio Caminha Muniz Neto
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

We start this chapter by inverting the viewpoint of Chap.  15 . More precisely, we fix a complex number z and examine the set of polynomials \(f\in \mathbb C[X]\) for which f(z) = 0. As a byproduct of our discussion, we give a (hopefully) more natural proof of the closedness, with respect to the usual arithmetic operations, of the set of complex numbers which are roots of nonzero polynomials of rational coefficients. We then proceed to investigate the special case of roots of unity, which leads us to the study of cyclotomic polynomials and allows us to give a partial proof of a famous theorem of Dirichlet on the infinitude of primes on certain arithmetic progressions. The chapter closes with a few remarks on the set of real numbers which are not roots of nonzero polynomials with rational coefficients.

References

  1. 6.
    R. Ash, Basic Abstract Algebra: for Graduate Students and Advances Undergraduates (Dover, Mineola, 2006)Google Scholar
  2. 7.
    A. Caminha, Uma prova elementar de que os números complexos algébricos sobre \(\mathbb Q\) formam um corpo. Matemática Universitária 52/53, 14–17 (2015) (in Portuguese)Google Scholar
  3. 8.
    A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)Google Scholar
  4. 12.
    R. Courant, H. Robbins, What Is Mathematics (Oxford University Press, Oxford, 1996)Google Scholar
  5. 17.
    D.G. de Figueiredo, Números Irracionais e Transcendentes (in Portuguese) (SBM, Rio de Janeiro, 2002)Google Scholar
  6. 20.
    C.R. Hadlock, Field Theory and its Classical Problems (Washington, MAA, 2000)Google Scholar
  7. 27.
    E. Landau, Elementary Number Theory (AMS, Providence, 1999)Google Scholar
  8. 28.
    S. Lang, Algebra (Springer, New York, 2002)Google Scholar
  9. 33.
    W. Rudin, Principles of Mathematical Analysis, 3rd edn. (McGraw-Hill, Inc., New York, 1976)Google Scholar
  10. 36.
    E. Stein, R. Shakarchi, Fourier Analysis: An Introduction (Princeton University Press, Princeton, 2003)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Antonio Caminha Muniz Neto
    • 1
  1. 1.Universidade Federal do CearáFortalezaBrazil

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