A table of the first factor for prime cyclotomic fields
A table of the first factor for prime cyclotomic fields
复制标题
素分圆域的第一个因子表
DOI：
10.1090/s0025-5718-1970-0257029-5
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发表时间：
1970
影响因子：
2
通讯作者：
M. Newman
中科院分区：
文献类型：
--
作者：
M. Newman
用行列式公式计算了所有素数200的素割圆域的第一因子，纠正了Kummer表中的一些错误。设p=2m+1为奇素数。设f是本原的第p个单位根，R是有理数域。众所周知，如果h是分圆域Are(F)的类号，H0是全实子域Are(f+1/f)的类号，则h可被H0整除。商h/ho用h*表示，被称为h的第一个因数。关于这些问题的完整讨论可以在Borevic和Safarevic写的漂亮的书[1]中找到，书中给出了所有奇素数p<；100的h*表(未记帐)。据推测，这个表是由于Kummer计算了所有奇素数^163的h*(见[2]和[3])。数字h*很难计算，验证和扩展上面提到的表是有意义的。H*的重要性源于这样一个事实：素数p是不规则的当且仅当p除以h*。事实证明，Rummer的表格并非没有错误：对应于p=103、139和163的h*值是不正确的。事实是，p=103是不规则的，库默正确地确认了它是不规则的，这是可以用他的计算方法解释的。设g是模p的本原根，定义9n=gnp[gn/p]，n=0，1，2，···设8是一个原始的(p-L)一元根。然后，第一个因子h*由公式1 7747/ys\7-7//.3\t4/qp-2\其中h*=--；F(0)F(Tf)···F(0)，(2p)m-1 V-2 Fix)=X)9“Xn
The first factor of the prime cyclotomic fields for all primes < 200 is computed by means of a determinantal formula, correcting some errors in tables of Kummer. Let p = 2m + 1 be an odd prime. Let f be a primitive pth root of unity, and let R be the field of rationals. It is well known that if h is the class number of the cyclotomic field Ä(f), and h0 the class number of the totally real subfield Ä(f + 1/f), then h is divisible by h0. The quotient h/ho is denoted by h*, and is known as the first factor of h. A complete discussion of these matters is to be found in the beautiful book [1] by Borevic and Safarevic, where a table (uncredited) of h* is given for all odd primes p < 100. Presumably, this table is due to Kummer, who computed h* for all odd primes ^ 163 (see [2] and [3]). The numbers h* are quite difficult to compute, and it is of some interest to verify and to extend the above-mentioned tables. The importance of h* stems from the fact that the prime p is irregular if and only if p divides h*. It turns out that Rummer's tables are not error-free: the values of h* corresponding to p = 103, 139, and 163 are incorrect. The fact that p = 103 is irregular, and was correctly identified to be so by Kummer, is explicable by his method of computation. Let g be a primitive root modulo p. Define 9n = gn p[gn/p] , n = 0, 1, 2, • • • . Let 8 be a primitive (p — l)st root of unity. Then the first factor h* is given by the formula 1 7747 /ys\ 7-7 //.3\ T4/Qp—2\ where h* = ——; F(0)F(tf) • • • F(0 ) , (2p)m-1 V—2 Fix) = X) 9"Xn ■