In algebraic number theory, the concept of principalization (also called capitulation) refers to the phenomenon where an ideal (or more generally a fractional ideal) of the ring of integers of a number field, which is not principal in that field, becomes principal after extension to the ring of integers of a larger algebraic number field. The study of principalization originates in the work of Ernst Kummer in the 1840s on ideal numbers. Kummer showed that for every algebraic number field there exists an extension in which all ideals of its ring of integers (which can always be generated by at most two elements) become principal. In 1897, David Hilbert conjectured that the Hilbert class field (the maximal abelian extension of a number field that is unramified everywhere) provides such an extension. This statement, now known as the principal ideal theorem, was proved in 1930 by Philipp Furtwängler, following its reformulation by Emil Artin in 1929 using his general reciprocity law. Furtwängler’s proof relied on Artin transfers in non-abelian groups of derived length two. Building on this, researchers sought to apply group-theoretic methods to study principalization in intermediate fields between a base field and its Hilbert class field. The first significant contributions were made in 1934 by Arnold Scholz and Olga Taussky, who introduced the synonym capitulation for principalization. An alternative approach to the principalization problem, based on Galois cohomology of unit groups, also goes back to Hilbert. In his Zahlbericht, he developed this perspective in the context of cyclic extensions of prime degree, culminating in the celebrated Hilbert’s Theorem 94.
Extension of classes Let K {\displaystyle K} be an algebraic number field, called the base field, and let L / K {\displaystyle L/K} be a field extension of finite degree. Let O K , I K , P K {\displaystyle {\mathcal {O}}_{K},{\mathcal {I}}_{K},{\mathcal {P}}_{K}} and O L , I L , P L {\displaystyle {\mathcal {O}}_{L},{\mathcal {I}}_{L},{\mathcal {P}}_{L}} denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields K , L {\displaystyle K,L} respectively. Then the extension map of fractional ideals
{ ι L / K : I K → I L a ↦ a O L {\displaystyle {\begin{cases}\iota _{L/K}:{\mathcal {I}}_{K}\to {\mathcal {I}}_{L}\\{\mathfrak {a}}\mapsto {\mathfrak {a}}{\mathcal {O}}_{L}\end{cases}}}
is an injective group homomorphism. Since ι L / K ( P K ) ⊆ P L {\displaystyle \iota _{L/K}({\mathcal {P}}_{K})\subseteq {\mathcal {P}}_{L}} , this map induces the extension homomorphism of ideal class groups
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