In the mathematical field of
algebraic number theory, the concept of
principalization refers to a situation when given an
extension of
algebraic number fields, some
ideal (or more generally
fractional ideal) of the
ring of integers of the smaller field isn't
principal but it's
extension to the ring of integers of the larger field is. Its study has origins in the work of
Ernst Kummer on
ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897
David Hilbert conjectured that
maximal abelian unramified extension of the base field, which was later called the
Hilbert class field of the given base field, is such an extension. This conjecture, now known as
principal ideal theorem, was proved by
Philipp Furtwängler in 1930 after it had been translated from
number theory to
group theory by
Emil Artin in 1929, who made use of his
general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of
Artin transfers of
non-abelian groups with
derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to
Arnold Scholz and
Olga Taussky in 1934, who coined the synonym
capitulation for principalization. Another independent access to the principalization problem via
Galois cohomology of
unit groups is also due to Hilbert and goes back to the chapter on
cyclic extensions of number fields of prime
degree in his
number report, which culminates in the famous
Theorem 94.