The subgroup of the ideal class group of a base global field consisting of those ideal classes that become principal in a specified finite extension; equivalently the kernel of the natural transfer (or capitulation) map from the class group of the base field to the class group of the extension.

Principle

Capitulation measures principalization under extension: a class capitulates exactly when its image under the transfer map vanishes, so the capitulation kernel organizes which global ideal-theoretic obstructions are removed by passing to the extension.

Demonstration

For the Hilbert class field H of a number field K every ideal class of K becomes principal in H, so the capitulation kernel for H/K is the whole class group Cl(K); in a cyclic degree-p extension L/K one can compute the capitulation kernel as the kernel of the norm-induced map on class groups and observe partial capitulation of a subgroup of Cl(K).

Misapplication

Confusing the capitulation kernel with the kernel of the norm map on fractional ideals or with the ambiguous class group; the capitulation kernel concerns ideal classes becoming principal in the extension, not merely classes fixed by Galois action or mapped trivially by norms on ideals.

Consequence

Identifying the capitulation kernel constrains class group growth and determines which class field theoretic obstructions disappear in the extension; it guides construction of further unramified extensions and informs the structure of the Galois module Cl(L).

Reversal

The complementary phenomenon is persistence: classes that remain nonprincipal in every extension of a given type, producing stable obstructions to principalization and reflecting deeper arithmetic invariants.

Boundary

Defined for finite extensions of global fields (number fields or global function fields) with respect to ideal class groups; it does not apply directly to local fields, to arbitrary Picard groups of schemes without translation to divisor classes, nor to contexts where no natural transfer map on class groups exists.

Semantic Tension

Close to the ambiguous class group (classes fixed by Galois) and to kernels of norm maps; the tension lies in principalization (capitulation) vs Galois-fixedness or norm-triviality, which coincide only in special situations.

Synthesis

The capitulation kernel is the subgroup of ideal classes eliminated by passing to a chosen extension; as the kernel of the class-group transfer map it encapsulates which global ideal-theoretic obstructions are removed by that extension and thereby links explicit class field constructions to the arithmetic of ideal principalization.