Skip to main content
Log in

Lindemann–Weierstrass Theorem

  • Published:
Moscow University Mathematics Bulletin Aims and scope


We discuss issues of algebraic independence for values of the function \(e^{z}\) at algebraic points. The most general result of this kind was established at the end of the 19th century and is called the Lindemann–Weierstrass theorem. This is historically the first theorem on the algebraic independence of numbers and it can be proved now in various ways. Below we propose one more way to prove it.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Ch. Hermite, Oeuvres de Charles Hermite (Gauthier–Villars, Paris, 1912).

    MATH  Google Scholar 

  2. F. Lindemann, ‘‘Ueber die Zahl \(\pi\),’’ Math. Ann. 20, 213–225 (1882).

    Article  MathSciNet  MATH  Google Scholar 

  3. K. Weierstrass, ‘‘Zu Lindemann’s Abhandlung: ‘Über die Ludolph’sche Zahl,’’’ (Sitzungsber. Preuss. Akad. Wiss., 1885), pp. 1067–1085.

    MATH  Google Scholar 

  4. E. V. Bedulev, ‘‘On the linear independence of numbers over number fields,’’ Math. Notes 64, 440–449 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  5. Yu. V. Nesterenko, ‘‘On linear independence of numbers,’’ Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 1, 46–49 (1985).

  6. T. Rivoal, ‘‘La fonction zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs,’’ C. R. Acad. Sci. A 331, 267–270 (2000).

    Article  MATH  Google Scholar 

  7. C. L. Siegel, ‘‘Uber einige Anvendungen diophantischer Approximationen,’’ Abh. Preuss. Akad. Wiss. Phys.-math. Kl., 1929, No. 1.

  8. K. Mahler, ‘‘Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I,’’ J. Reine Angew. Math. 166, 118–136 (1931).

    Article  MathSciNet  MATH  Google Scholar 

  9. ‘‘Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II,’’ J. Reine Angew. Math. 166, 137–150 (1931).

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yu. V. Nesterenko.

Ethics declarations

The author declares that he has no conflicts of interest.

Additional information

Translated by E. Oborin

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nesterenko, Y.V. Lindemann–Weierstrass Theorem. Moscow Univ. Math. Bull. 76, 239–243 (2021).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: