Heinrich Weber

Heinrich Weber (1842–1913) was a German mathematician whose work on algebraic number theory provided the essential scaffolding upon which Teiji Takagi and Emil Artin later built class field theory. His three-volume Lehrbuch der Algebra became the standard reference for a generation, and his development of the Weber Class — a generalization of the ideal class group — contained the seeds of what would become the full Takagi Existence Theorem. Weber's work on the Kronecker-Weber Theorem established the first complete result in class field theory, proving that every abelian extension of the rational numbers is contained in a cyclotomic field.

Weber is the forgotten predecessor: his theorems were correct but his proofs incomplete, his vision clear but his tools insufficient. The tragedy of his position is that he saw the landscape but lacked the altitude to traverse it. Takagi supplied the altitude.