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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
MathSciNet review:
0404204
Full-text PDF Free Access
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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
(19,533d)
- [3]
K.
K. Billevič, Letter to the editor, Mat. Sb. (N.S.)
48 (49) (1959), 256 (Russian). MR 0123554
(23 #A879)
- [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
(29 #1201)
- [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, 1966. MR 0195803
(33 #4001)
- [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 (2,349d)
- [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
(14,247d)
- [8]
Ove
Hemer, Notes on the Diophantine equation
𝑦²-𝑘=𝑥³, Ark. Mat.
3 (1954), 67–77. MR 0061115
(15,776h)
- [9]
Hymie
London and Raphael
Finkelstein, On Mordell’s equation
𝑦²-𝑘=𝑥³, Bowling Green State
University, Bowling Green, Ohio, 1973. MR 0340172
(49 #4928)
- [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 (17,240e)
- [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.
- [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 degrees," Mat. Sb., v. 40 (82), 1956, pp. 123-136. (Russian) MR 19, 533. MR 0088516 (19:533d)
- [3]
- K. K. BILLEVIČ, "Letter to the Editor," Mat. Sb., v. 48 (90), 1959, p. 256. (Russian) MR 23 #A879. MR 0123554 (23:A879)
- [4]
- K. K. BILLEVIČ, "A theorem on unit elements of algebraic fields of order n," Mat. Sb., v. 64 (106), 1962, pp. 145-152. (Russian) MR 29 #1201. MR 0163902 (29:1201)
- [5]
- Z. I. BOREVIČ & I. R. ŠAFAREVIČ, Number Theory, "Nauka", Moscow, 1964; English transl., Pure and Appl. Math., vol. 20, Academic Press, New York, 1966. MR 30 #1080; 33 #4001. MR 0195803 (33:4001)
- [6]
- B. N. DELONE & D. K. FADDEEV, The Theory of Irrationalities of the Third Degree, Trudy Mat. Inst. Steklov., v. 11, 1940; English transl., Transl. Math. Monographs, vol. 10, Amer. Math. Soc., Providence, R. I., 1964. MR 2, 349; 28 #3955. MR 0004269 (2:349d)
- [7]
- O. HEMER, On the Diophantine Equation
, Doctoral Dissertation, Uppsala, 1952. MR 0049917 (14:247d)
- [8]
- O. HEMER, "Notes on the Diophantine equation
," Ark. Mat., v. 3, 1954, pp. 67-77. MR 15, 776. MR 0061115 (15:776h)
- [9]
- H. LONDON & R. FINKELSTEIN, On Mordell's Equation
, Bowling Green State Univ., Bowling Green, Ohio, 1973. MR 49 #4928. MR 0340172 (49:4928)
- [10]
- H. B. MANN, Introduction to Algebraic Number Theory, Ohio State Univ. Press, Columbus, Ohio, 1956. MR 17, 240. MR 0072174 (17:240e)
- [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:
http://dx.doi.org/10.1090/S0025-5718-1976-0404204-8
PII:
S 0025-5718(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
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