The List of Hilbert’s Twenty-Three Problems: Hilbert’s Ninth Problem

Reciprocity Laws and Algebraic Number Fields: Find the most general law of the reciprocity theorem in any algebraic number field.

Hilbert’s ninth problem is on algebraic number fields, extensions of the rational numbers to include, say, √2 or certain complex numbers. Hilbert asked for the most general form of a reciprocity law in any algebraic number field, that is, the conditions that determine which polynomials can be solved within the number field. Partial solutions by Emil Artin, Teiji Takagi and Helmut Hasse have pushed the field further, although the question has not been answered in full. The closely related 12th problem, which deals with other extensions of the rational numbers, is unresolved.