On Euclid's Algorithm in cubic self-conjugate fields

H. Heilbronn

doi:10.1017/s0305004100025883

On Euclid's Algorithm in cubic self-conjugate fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100025883 ◽

1950 ◽

Vol 46 (3) ◽

pp. 377-382 ◽

Cited By ~ 2

Author(s):

H. Heilbronn

Keyword(s):

Finite Number ◽

Quadratic Fields ◽

Euclid’S Algorithm ◽

Euclid's Algorithm

In a paper published in these Proceedings I proved that there are only a finite number of quadratic fields in which Euclid's Algorithm (E.A.) holds. Recently Davenport has found a new proof of this theorem based on the theory of the minima of the product of linear inhomogeneous forms.

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On Euclid's Algorithm in Cyclic Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1951-029-4 ◽

1951 ◽

Vol 3 ◽

pp. 257-268 ◽

Cited By ~ 3

Author(s):

H. Heilbronn

Keyword(s):

Finite Number ◽

Quadratic Fields ◽

Euclid’S Algorithm ◽

Cubic Fields ◽

Euclid's Algorithm ◽

Totally Real ◽

Cyclic Cubic Fields

In two papers I have proved that there are only a finite number of quadratic fields [6] and of cyclic cubic fields [7] in which Euclid's algorithm (E.A.) holds. Davenport has shown by a different method that there are only a finite number of quadratic fields [1, 2], of non-totally real cubic fields [3, 4] and of totally complex quartic fields in which E.A. holds.

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Euclid's Algorithm in real Quadratic Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1950-026-7 ◽

1950 ◽

Vol 2 ◽

pp. 289-296 ◽

Cited By ~ 10

Author(s):

H. Chatland ◽

H. Davenport

Keyword(s):

Quadratic Fields ◽

Real Quadratic Fields ◽

Euclid’S Algorithm ◽

Integral Element ◽

Euclid's Algorithm ◽

Real Quadratic

1. Let m be a positive square-free integer. Euclid's Algorithm is said to hold in the field if, given any non-integral element a in the field, an integer a can be found so that

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Indefinite Binary Quadratic Forms, and Euclid's Algorithm in Real Quadratic Fields

Proceedings of the London Mathematical Society ◽

10.1112/plms/s2-53.2.81 ◽

1951 ◽

Vol s2-53 (1) ◽

pp. 81-82

Author(s):

H. Davenport

Keyword(s):

Quadratic Forms ◽

Quadratic Fields ◽

Binary Quadratic Forms ◽

Real Quadratic Fields ◽

Euclid’S Algorithm ◽

Euclid's Algorithm ◽

Real Quadratic

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Indefinite Binary Quadratic Forms, and Euclid's Algorithm in Real Quadratic Fields

Proceedings of the London Mathematical Society ◽

10.1112/plms/s2-53.1.65 ◽

1951 ◽

Vol s2-53 (1) ◽

pp. 65-82 ◽

Cited By ~ 10

Author(s):

H. Davenport

Keyword(s):

Quadratic Forms ◽

Quadratic Fields ◽

Binary Quadratic Forms ◽

Real Quadratic Fields ◽

Euclid’S Algorithm ◽

Euclid's Algorithm ◽

Real Quadratic

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On Euclid's algorithm in real quadratic fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410002051x ◽

1938 ◽

Vol 34 (4) ◽

pp. 521-526 ◽

Cited By ~ 4

Author(s):

H. Heilbronn

Keyword(s):

Quadratic Field ◽

Quadratic Fields ◽

Real Quadratic Fields ◽

Euclid’S Algorithm ◽

Euclid's Algorithm ◽

Real Quadratic

The object of this paper is to complete the proof of theTheorem.LetP(√d)be the quadratic field of discriminant d> 0.Then Euclid's algorithm does not hold inP(√d)if d is sufficiently large.

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