Artin's early work was concentrated on the analytical and arithmetical theory of quadratic number fields. In his doctoral thesis of 1921 he formulated the analogue of the Riemann hypothesis about the zeros of the classical zeta function, studying the quadratic extension of the field of rational functions of one variable over finite constant fields, by applying the arithmetical and analytical theory of quadratic numbers over the field of natural numbers. Then in 1923, in the most important discovery of his career, he derived a functional equation for his new-type L-series. The proof of this he published in 1927, thereby providing, by the use of the theory of formal real fields, the solution to Hilbert's problem of definite functions. The proof produced the general law of reciprocity - Artin's phrase - which included all previously known laws of reciprocity and became the fundamental theorem in class field theory.

Artin made two other important theoretical advances. His theory of braids, given in 1925, was a major contribution to the study of nodes in three-dimensional space. A year later, in collaboration with Schrier, he succeeded in treating real algebra in an abstract manner, defining a field as real-closed if it itself was real but none of its algebraic extensions were. He was then able to demonstrate that a real-closed field could be ordered in an exact manner and that, in it, typical laws of algebra were valid.

In 1944 his discovery of rings with minimum conditions for right ideals - now known as Artin rings - was a fertile addition to the theory of associative ring algebras.

Class Field Theory (1961) is a summation of his life's work.

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Artin, Emil