A Table of Totally Real Cubic Fields

Angell, I. O.

doi:10.2307/2005443

Cited by 10 publications

(32 citation statements)

References 4 publications

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“…Using similar methods Angelí [1] extends these results up to D < 100000. Using a different method the authors [5] have constructed a table of the 4753 real nonabelian cubic fields with discriminants D < 100000.…”

supporting

confidence: 54%

On the real cubic fields

Llorente¹,

Oneto²

1982

Math. Comp.

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supporting

confidence: 54%

On the real cubic fields

Llorente¹,

Oneto²

1982

Math. Comp.

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“…It is therefore clear that these complex cubic fields must have a class number divisible by 3. Table III 315 A comparison with Angell's table [1] of complex cubic fields of discriminant larger than -(4 • IO9)1/3 and a class number divisible by 3 shows that only the three nonisomorphic fields of discriminant -1228 do not appear in our Table III. It is nonetheless easy to verify that the ninth-degree fields containing such a cubic subfield are outside the limits of table (3, 3)c • When the real cubic subfield k of K (with the notations in Table III) is abelian, K is a relative cyclic extension of the complex cubic field k'.…”

Section: Description Of the Tables And Their Computing Timementioning

confidence: 88%

“…j=x The most tedious case is the totally real one, where this test may require 64 = 1296 trials; the easiest case corresponds to the signature (1,4), where at most 24 trials are necessary. Once a suitable permutation for the roots is obtained, it is easy to compute the equations with rational coefficients (the norm of the index [ZK : Z[0]] is the denominator of the coefficients) relating the roots of Q(X) and the roots of P(X).…”

Section: Relative Cubic Extensions Of a Cubic Fieldmentioning

confidence: 99%

“…The Galois group of the fields in the tables are, according to the signature: (9, 0) : two fields of type Tx+ = C9 , one of type T2 = C2 , one of type Ti, = C3 x S3, two of type Txl, one of type T20 , 19 of type T2$ and one of type Tix • (7,1) : all the fields of type T2%. (5,2): 148 fields of type T2% and 6 of type T3i.…”

Section: Galois Groupsmentioning

confidence: 99%

“…For this reason, extensive tables of number fields exist only for degrees up to six ( [1,2,5,6,10,11,12,14,19,21,22,23,24,29]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Imprimitive ninth-degree number fields with small discriminants

Diaz¹,

Olivier²

1995

Math. Comp.

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Abstract. We present tables of ninth-degree nonprimitive (i.e., containing a cubic subfield) number fields. Each table corresponds to one signature, except for fields with signature (3,3), for which we give two dilferent tables depending on the signature of the cubic subfield. Details related to the computation of the tables are given, as well as information about the CPU time used, the number of polynomials that we deal with, etc. For each field in the tables, we give its discriminant, the discriminant of its cubic subfields, the relative polynomial generating the field over one of its cubic subfields, the corresponding (irreducible) polynomial over Q, and the Galois group of the Galois closure. Fields having interesting properties are studied in more detail, especially those associated with sextic number fields having a class group divisible by 3.

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Finding fundamental units in cubic fields

Cusick

1982

Math. Proc. Camb. Phil. Soc.

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This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

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A Table of Totally Real Cubic Fields

Cited by 10 publications

References 4 publications

On the real cubic fields

On the real cubic fields

Imprimitive ninth-degree number fields with small discriminants

Finding fundamental units in cubic fields

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