On an algorithm of Billevich for finding units in algebraic fields

Authors:
Ray Steiner and Ronald Rudman

Journal:
Math. Comp. **30** (1976), 598-609

MSC:
Primary 12A45

DOI:
https://doi.org/10.1090/S0025-5718-1976-0404204-8

MathSciNet review:
0404204

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Abstract | References | Similar Articles | Additional Information

Abstract: The well-known algorithm of Billevich for finding units in algebraic number fields is derived by algebraic methods. Some tables of units in cubic and quartic fields are given.

**[1]**W. E. H. BERWICK, "Algebraic number fields with two independent units,"*Proc. London Math. Soc.*, v. 34 (2), 1932, pp. 360-378.**[2]**K. K. Billevič,*On units of algebraic fields of third and fourth degree*, Mat. Sb. N.S.**40(82)**(1956), 123–136 (Russian). MR**0088516****[3]**K. K. Billevič,*Letter to the editor*, Mat. Sb. (N.S.)**48 (49)**(1959), 256 (Russian). MR**0123554****[4]**K. K. Billevič,*A theorem on unit elements of algebraic fields of order 𝑛*, Mat. Sb. (N.S.)**64 (106)**(1964), 145–152 (Russian). MR**0163902****[5]**A. I. Borevich and I. R. Shafarevich,*Number theory*, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR**0195803****[6]**B. N. Delone and D. K. Faddeev,*Theory of Irrationalities of Third Degree*, Acad. Sci. URSS. Trav. Inst. Math. Stekloff,**11**(1940), 340 (Russian). MR**0004269****[7]**Ove Hemer,*On the solvability of the Diophantine equation 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²=0 in imaginary Euclidean quadratic fields*, Ark. Mat.**2**(1952), 57–82. MR**0049917**, https://doi.org/10.1007/BF02591382**[8]**Ove Hemer,*Notes on the Diophantine equation 𝑦²-𝑘=𝑥³*, Ark. Mat.**3**(1954), 67–77. MR**0061115**, https://doi.org/10.1007/BF02589282**[9]**Hymie London and Raphael Finkelstein,*On Mordell’s equation 𝑦²-𝑘=𝑥³*, Bowling Green State University, Bowling Green, Ohio, 1973. MR**0340172****[10]**Henry B. Mann,*Introduction to algebraic number theory*, The Ohio State University Press, Columbus, Ohio, 1955. With a chapter by Marshall Hall, Jr. MR**0072174****[11]**G. F. VORONOĬ,*On a Generalization of the Algorithm of Continued Fractions*, Doctoral Dissertation, Warsaw, 1896. (Russian)**[12]**H. C. WILLIAMS & C. R. ZARNKE,*A Table of Fundamental Units for Cubic Fields*, Scientific Report No. 63, University of Manitoba, 1973.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0404204-8

Keywords:
Algebraic number field,
units,
Billevich's algorithm,
multiplicative lattices,
Cramer's rule,
fundamental units

Article copyright:
© Copyright 1976
American Mathematical Society