In mathematics, the Zahlbericht (number report) was a report on algebraic number theory

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

 by .

History

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

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

 (republished in 1 volume as ).

In 1893 the German mathematical society invited Hilbert and Minkowski

Hermann Minkowski

Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

 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.

Contents

Part 1 covers the theory of general number fields, including ideals, discriminant

Discriminant

In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

s, differents, units, and ideal classes.

Part 2 covers Galois number field

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

s, including in particular Hilbert's theorem 90

Hilbert's Theorem 90

In abstract algebra, Hilbert's Theorem 90 refers to an important result on cyclic extensions of fields that leads to Kummer theory...

.

Part 3 covers quadratic number fields, including the theory of genera, and class number

Class number

In mathematics, class number may refer to* Class number , the number of conjugacy classes of a group*The size of the ideal class group of a number ring*The number of equivalence classes of binary quadratic forms of a given discriminant...

s of quadratic fields.

Part 4 covers cyclotomic field

Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...

s, including the Kronecker–Weber theorem

Kronecker–Weber theorem

In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...

 (theorem 131), the Hilbert–Speiser theorem (theorem 132), and the Eisenstein reciprocity law for lth power residues (theorem 140) .

Part 5 covers Kummer number fields, and ends with Kummer's proof of Fermat's last theorem for regular primes.

External links

• Introduction to the English Edition of Hilbert's Zahlbericht