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On a Theorem of Liouville in Fields of Positive Characteristic

Published online by Cambridge University Press:  20 November 2018

K. Mahler
K. Mahler*
Affiliation:
Institute for Advanced Study, PrincetonN.J.
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A classical theorem of J. Liouville states that if z is a real algebraic number of degree n ≥ 2, then there exists a constant c > 0 such that for every pair of integers a, b with b ≠ 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 4 , 01 August 1949 , pp. 397 - 400
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 C.R. Acad. Sci. Paris, vol. 18 (1844), 883-885, 910-911.

2 Norske Vid. Selsk. Scr.(1908), Nr. 7.

3 Math. ZeiL, vol. 10 (1921), 173-213.

4 Acta Math., vol. 79 (1947), 225-240.

5 Ann. of Math. (2) 31 (1930), 207-218.

6 I am indebted to E. Artin for the remark that z is algebraic if k is of characteristic p. If k is of characteristic 0, then z is, of course, transcendental over k(x).