Class numbers of cyclotomic fields

Cornell, Gary; Washington, Lawrence C.

doi:10.1016/0022-314x(85)90055-1

Cited by 38 publications

(36 citation statements)

References 6 publications

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“…We also assume that Q > 2c2. We then have We will do this by extending the method of Cornell and Washington [5]. For this purpose we let G be the …”

confidence: 99%

On the Computation of the Class Number of an Algebraic Number Field

Buchmann¹,

Williams²

1989

Mathematics of Computation

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Abstract.It is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field F. The method assumes the truth of the Generalized Riemann Hypothesis in order to estimate the residue of the Dedekind zeta function of F at s = 1 sufficiently well that h can be determined unambiguously. Given the regulator R of F and a known divisor h* of h, it is shown that this technique will produce the value of h in 0(\dp\1+e/(h*R)2) elementary operations, where djr is the discriminant of F. Thus, if h < Idpl1/8, then the complexity of computing h (with h* = 1) is 0(|df \l'4+£).

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“…We also assume that Q > 2c2. We then have We will do this by extending the method of Cornell and Washington [5]. For this purpose we let G be the …”

confidence: 99%

On the Computation of the Class Number of an Algebraic Number Field

Buchmann¹,

Williams²

1989

Mathematics of Computation

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“…It is amusing to note that we encounter the equation 2wA = y2 + 1 considered by Ljunggren [9,14], and its (perhaps unexpected) integer point (w, y) = (13,239). One reason for studying the "simplest fields" is that the roots of the polynomials yield explicit units that often generate subgroups of small index in the full groups of units [2,3,4,5,6,7,10,11,12,13]. In the present situation, since the extensions we obtain are non-Galois and therefore contain few roots, we are forced to consider the Galois closure, in which case we cannot hope to obtain a maximal set of independent units as roots of our polynomials.…”

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confidence: 99%

A family of real 2ⁿ-tic fields

Shen¹,

Washington²

1994

Trans. Amer. Math. Soc.

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Abstract. We study the family of polynomials P"(X; a) = n((* + if) -~nX + i)2") and determine when P"(X ; a), a € Z , is irreducible. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the 2"th cyclotomic field. The families of fields we obtain are natural extensions of those studied by M.-N. Gras and Y.-Y. Shen, but in general the present fields are non-Galois for n > 4. From the roots we obtain a set of independent units for the Galois closure that generate an "almost fundamental piece" of the full group of units. Finally, we discuss the two examples where our fields are Galois, namely a = ±2" and a = ±24 • 239.

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“…These class numbers have been computed in [7] for the cubic, and in [4] for the quartic case. A simplest cubic field is used in [6] to compute the first example where the real part of the class number of the pth cyclotomic field is larger than p; similarly, the corresponding quartic fields are used in [1] to prove a general inequality for the real part of the class number of some cyclotomic fields. Real cyclic sextic fields are studied in [5], where a table of units and class numbers (for conductors less than 2021) is computed.…”

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confidence: 99%