In
mathematics, the ''Zahlbericht'' (number report) was a report on
algebraic number theory by .
History
In 1893 the German mathematical society invited Hilbert and
Minkowski to write reports on the theory of numbers. They agreed that Minkowski would cover the more elementary parts of number theory while Hilbert would cover algebraic number theory. Minkowski eventually abandoned his report, while Hilbert's report was published in 1897. It was reprinted in volume 1 of his collected works, and republished in an English translation in 1998.
and and the English introduction to give detailed discussions of the history and influence of Hilbert's ''Zahlbericht''.
Some earlier reports on number theory include the report by
H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in , and the report by . wrote an update of Hilbert's ''Zahlbericht'' that covered
class field theory (republished in 1 volume as ).
Contents
Part 1 covers the theory of general
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s, including ideals,
discriminants, differents, units, and ideal classes.
Part 2 covers
Galois number fields, including in particular
Hilbert's theorem 90
In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is an extension of ...
.
Part 3 covers
quadratic number fields, including the theory of genera, and
class numbers of quadratic fields.
Part 4 covers
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
s, including the
Kronecker–Weber theorem
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conve ...
(theorem 131), the
Hilbert–Speiser theorem (theorem 132), and the
Eisenstein reciprocity law for ''l''th power residues (theorem 140) .
Part 5 covers
Kummer number fields, and ends with Kummer's proof of Fermat's last theorem for
regular prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli nu ...
s.
References
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External links
{{wikisource, de:David Hilbert Gesammelte Abhandlungen Erster Band – Zahlentheorie/Kapitel 7, Die Theorie der algebraischen Zahlkörper.
Introduction to the English Edition of Hilbert's Zahlbericht
1897 in science
1897 non-fiction books
Algebraic number theory
History of mathematics
Mathematics books
Treatises