It is shown how the analytic class number formula can be used to produce ... estimate the residue of the Dedekind zeta function of F at s = 1 sufficiently well that h can be ... to compute large tables of quadratic and cubic class numbers. (See ... whether this method can be used to determine the class number of an arbitrary.
MATHEMATICS OF COMPUTATION VOLUME 53, NUMBER 188 OCTOBER 1989, PAGES 679-688
On the Computation of the Class Number
of an Algebraic Number Field By Johannes
Buchmann
and H. C. Williams*
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+£).
1. Introduction. Let F be an algebraic number field of degree n over the rationals Q. It is well known that the class number h of F can be expressed by means of the analytic class number formula
(1.1)
h = CFlim(s-l)çF(s) S—tl
with
cF
ii,i/\dc 0 such that n-1
n-1
k+2 Y, ak(p)P~
5>(p)*r*+1
< Cx and
Q. We also assume that Q > 2c2. We then have 1 n—l
\ak(p)\
2—1^2 2-é
p>Q r
k=2
^
k-2
Qj=2
1. In this case, itj has no fixed point. Also,
(2.4)
£,(p) 3 ' =-Í---f-. (pf^-1 + pf»-2 + ■■■+ l)(p/a> - 1) • • • (ph>'> - 1)
Since p is assumed to be unramified, we have Y^iLi fij = n; hence, upon multiplying out the denominator of (2.4) we see that the coefficient of pn~2 is precisely 1. Case 2. fxj = 1. In this case it is easy to see that —axj is the number of values
of i for which fij = 1 (2 < i < kj). It follows that axj = 1 —Nj. PROPOSITION 2.2.
D
For the values of m and axj defined above we have m
5>ulc,-|= o. 3= 1
Proof. For a permutation -k E G denote by 9(tt) the number of fixed points of it. By Proposition 2.1 we have m
X>i¿|C;| = |G|- X>(tt). 3= 1
ir€G
But by Burnside's Orbit Lemma (see Grove [9]) we know that Z^eG^M and this proves our assertion. D Also, we note that since pn-l
E(p) >- -:-^pn-l nn —1 +pn-¿ _i_ r,n-2
„_1
H-1_ i
I i 1
>
we have
(2.5)
F^XlogQ)-1
by Mertens' Theorem (see Hardy and Wright [10, p. 351]). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
= M>
THE CLASS NUMBER OF AN ALGEBRAIC NUMBER FIELD
683
3. Determination of the Value of Q. In order to find the value of Q needed to correctly determine h, we define as in [5] the function Q d > 1 as described in Knuth [11, Algorithm D, p. 429]. This algorithm will execute in a time interval which is polynomial in logp (and n). If p does divide /, then the method of Buchmann and Lenstra [4] can be used to determine the decomposition type of p. This algorithm also requires a polynomial function of logp (and n) elementary operations in order to successfully terminate. We also need to analyze the total complexity of this algorithm. As this will depend on the choice of the defining polynomial /( 0 there exists a number K such that K = 0(\dF\1+*/{h*R)2)
and (6.1) holds for any Q > K. Thus, we have proved
THEOREM 6.1.
Algorithm 4.1 computes the class number h of F in 0(\dp\1+£/(h*R)2)
elementary operations.
If h< \dF\a, then by (3.2) we have Ä>(|dF|1/2"a+£);
by Theorem
6.1 we can compute
h (using h* = 1) in 0(\dF\2a+e)
operations.
Thus, Algorithm 4.1 will find h quite quickly when h is small, say h < {dp]1/8. For large values of h it is necessary first to determine a value for h* which will render the quantity \dF\/(h*R)2 sufficiently small that Algorithm 4.1 can efficiently com-
pute H. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
JOHANNES BUCHMANN AND H. C. WILLIAMS
688
7. Acknowledgments. The authors gratefully acknowledge several helpful suggestions concerning this work from Larry Grove and Richard Blecksmith. Mathematisches Institut Universität Düsseldorf 4000 Düsseldorf Federal Republic of Germany Department of Computer University of Manitoba
Science
Winnipeg R3T 2N2, Manitoba Canada
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algebraischer
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