Use of a computer scan to prove and are Euclidean

Authors:
Harvey Cohn and Jesse Deutsch

Journal:
Math. Comp. **46** (1986), 295-299

MSC:
Primary 11R16; Secondary 11H50, 11Y40

MathSciNet review:
815850

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The fields in the title are shown to be norm-Euclidean by a computer scan of the unit 4-cube representing coordinates of a field element translated by integers. The method is to subdivide this cube into sufficiently many small boxes so the norm is less than unity in each box, when referred to an appropriate "neighboring" integer.

**[1]**Helmut Bauer,*Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper*, J. Number Theory**1**(1969), 161–162 (German, with English summary). MR**0240072****[2]**Harvey Cohn,*A numerical study of Weber’s real class number of calculation. I*, Numer. Math.**2**(1960), 347–362. MR**0122809****[3]**Harvey Cohn,*A classical invitation to algebraic numbers and class fields*, Springer-Verlag, New York-Heidelberg, 1978. With two appendices by Olga Taussky: “Artin’s 1932 Göttingen lectures on class field theory” and “Connections between algebraic number theory and integral matrices”; Universitext. MR**506156****[4]**H. J. Godwin,*Real quartic fields with small discriminant*, J. London Math. Soc.**31**(1956), 478–485. MR**0082526****[5]**H. J. Godwin,*On Euclid’s algorithm in some quartic and quintic fields*, J. London Math. Soc.**40**(1965), 699–704. MR**0184928****[6]**H. W. Lenstra Jr.,*Euclidean number fields of large degree*, Invent. Math.**38**(1976/77), no. 3, 237–254. MR**0429826****[7]**Hendrik W. Lenstra Jr.,*Euclidean number fields. II, III*, Math. Intelligencer**2**(1979/80), no. 2, 73–77, 99–103. Translated from the Dutch by A. J. Van der Poorten. MR**577555**, 10.1007/BF03023376**[8]**John Myron Masley,*Class numbers of real cyclic number fields with small conductor*, Compositio Math.**37**(1978), no. 3, 297–319. MR**511747**

Retrieve articles in *Mathematics of Computation*
with MSC:
11R16,
11H50,
11Y40

Retrieve articles in all journals with MSC: 11R16, 11H50, 11Y40

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815850-8

Keywords:
Euclidean algorithm

Article copyright:
© Copyright 1986
American Mathematical Society