A method for computing the Iwasawa -invariant

Authors:
R. Ernvall and T. Metsänkylä

Journal:
Math. Comp. **49** (1987), 281-294

MSC:
Primary 11R29; Secondary 11R20, 11R23, 11Y40

DOI:
https://doi.org/10.1090/S0025-5718-1987-0890270-X

MathSciNet review:
890270

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Abstract: We present a method for computing the minus-part of the Iwasawa -invariant of an Abelian field *K*. Applying this method, we have computed for several odd primes *p* when *K* runs through a large number of quadratic extensions of the *p*th cyclotomic field. A report on these computations and an analysis of the results is included.

**[1]**R. Ernvall, "Generalized Bernoulli numbers, generalized irregular primes, and class number,"*Ann. Univ. Turku. Ser. A I*, No. 178, 1979, 72 pp. MR**533377 (80m:12002)****[2]**R. Ernvall & T. Metsänkylä, "Cyclotomic invariants and*E*-irregular primes,"*Math. Comp.*, v. 32, 1978, pp. 617-629. MR**482273 (80c:12004a)****[3]**R. Gold, "Examples of Iwasawa invariants,"*Acta Arith.*v. 26, 1974, pp. 21-32. MR**0347771 (50:272)****[4]**R. Gold, "Examples of Iwasawa invariants, II,"*Acta Arith.*, v. 26, 1975, pp. 232-240. MR**0374085 (51:10285)****[5]**F. H. Hao & C. J. Parry, "Generalized Bernoulli numbers and*m*-regular primes,"*Math. Comp.*, v. 43, 1984, pp. 273-288. MR**744938 (85h:11062)****[6]**S. Kobayashi, "Calcul approché de la série d'Iwasawa pour les corps quadratiques ,"*Number Theory*, 1981-82 and 1982-83, Exp. No. 4, 68 pp., Publ. Math. Fac. Sci. Besançon, Univ. Franche-Comté, Besançon, 1983.**[7]**H. W. Leopoldt, "Eine Verallgemeinerung der Bernoullischen Zahlen,"*Abh. Math. Sem. Univ. Hamburg*, v. 22, 1958, pp. 131-140. MR**0092812 (19:1161e)****[8]**T. Metsänkylä, "Iwasawa invariants and Kummer congruences,"*J. Number Theory*, v. 10, 1978, pp. 510-522. MR**515058 (80d:12007)****[9]**I. Sh. Slavutskii, "Local properties of Bernoulli numbers and a generalization of the Kummer-Vandiver theorem,"*Izv. Vyssh. Uchebn. Zaved. Mat.*, No. 3 (118), 1972, pp. 61-69. (Russian) MR**0300991 (46:151)****[10]**S. S. Wagstaff, Jr., "The irregular primes to 125000,"*Math. Comp.*, v. 32, 1978, pp. 583-591. MR**0491465 (58:10711)****[11]**L. C. Washington,*Introduction to Cyclotomic Fields*, Springer-Verlag, Berlin and New York, 1982. MR**718674 (85g:11001)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0890270-X

Article copyright:
© Copyright 1987
American Mathematical Society