Cyclotomic Fields I And Ii (graduate Texts In Mathematics) (v. 1-2)
by Serge Lang /
1989 / English / PDF
23.1 MB Download
Kummer's work on cyclotomic fields paved the way for the
development of algebraic number theory in general by Dedekind,
Weber, Hensel, Hilbert, Takagi, Artin and others. However, the
success of this general theory has tended to obscure special facts
proved by Kummer about cyclotomic fields which lie deeper than the
general theory. For a long period in the 20th century this aspect
of Kummer's work seems to have been largely forgotten, except for a
few papers, among which are those by Pollaczek [Po], Artin-Hasse
[A-H] and Vandiver [Va]. In the mid 1950's, the theory of
cyclotomic fields was taken up again by Iwasawa and Leopoldt.
Iwasawa viewed cyclotomic fields as being analogues for number
fields of the constant field extensions of algebraic geometry, and
wrote a great sequence of papers investigating towers of cyclotomic
fields, and more generally, Galois extensions of number fields
whose Galois group is isomorphic to the additive group of p-adic
integers. Leopoldt concentrated on a fixed cyclotomic field, and
established various p-adic analogues of the classical complex
analytic class number formulas. In particular, this led him to
introduce, with Kubota, p-adic analogues of the complex L-functions
attached to cyclotomic extensions of the rationals. Finally, in the
late 1960's, Iwasawa [Iw 11] made the fundamental discovery that
there was a close connection between his work on towers of
cyclotomic fields and these p-adic L-functions of Leopoldt -
Kubota.
Kummer's work on cyclotomic fields paved the way for the
development of algebraic number theory in general by Dedekind,
Weber, Hensel, Hilbert, Takagi, Artin and others. However, the
success of this general theory has tended to obscure special facts
proved by Kummer about cyclotomic fields which lie deeper than the
general theory. For a long period in the 20th century this aspect
of Kummer's work seems to have been largely forgotten, except for a
few papers, among which are those by Pollaczek [Po], Artin-Hasse
[A-H] and Vandiver [Va]. In the mid 1950's, the theory of
cyclotomic fields was taken up again by Iwasawa and Leopoldt.
Iwasawa viewed cyclotomic fields as being analogues for number
fields of the constant field extensions of algebraic geometry, and
wrote a great sequence of papers investigating towers of cyclotomic
fields, and more generally, Galois extensions of number fields
whose Galois group is isomorphic to the additive group of p-adic
integers. Leopoldt concentrated on a fixed cyclotomic field, and
established various p-adic analogues of the classical complex
analytic class number formulas. In particular, this led him to
introduce, with Kubota, p-adic analogues of the complex L-functions
attached to cyclotomic extensions of the rationals. Finally, in the
late 1960's, Iwasawa [Iw 11] made the fundamental discovery that
there was a close connection between his work on towers of
cyclotomic fields and these p-adic L-functions of Leopoldt -
Kubota.
