**Artin** made a major contribution to the theory of noncommutative rings and later worked on rings with the minimum condition on right ideals, now called Artinian rings.

- All his life Emil would have a love of music which essentially equalled his love of mathematics.
- Although the town today is called Liberec, and is in the northern Czech Republic, at the time that Emil was educated there it was a mainly German speaking city.
- However Artin did begin his university career, enrolling at the University of Vienna.
- At Hamburg Artin lectured on a wide variety of topics including mathematics, mechanics and relativity.
- These were particularly productive years for Artin's research.
- He made a major contribution to field theory, the theory of braids and, around 1928, he worked on rings with the minimum condition on right ideals, now called Artinian rings.
- It developed rapidly in the following decade and when Artin solved the following problem in 1924 he was following the natural progression for the topic.
- In his 1924 attack on this problem Artin restricted himself to considering only fields which were an algebraic closure of the field of rationals.
- In 1926 Artin published an important paper on joint work with Otto Schreier and we give some details below.
- Before looking further at the joint 1926 paper of Artin and Schreier we note that the pair published a 1927 paper in which they were able to handle the problem described above in the case of fields of prime characteristic.
- In this 1927 work they introduced what are called today Artin-Schreier cyclic extensions of degree ppp.
- The earlier research by Artin and Schreier had led them to define what today are called formally real fields, they are fields with the property that -1 cannot be expressed as a sum of squares.
- Artin himself proved that when OOO is the field of algebraic numbers, the subfield KKK of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of the field OOO.
- Artin and Schreier published in their famous 1926 paper their studies of all formally real fields and real closed fields, showing that a specific ordering could be defined on them.
- Now that the connection had been made with ordered fields, Artin was able to apply these methods to solve Hilbert's 17th problem.
- Artin gave a complete solution in the paper Über die Zerlegung definiter Funktionen in Quadrate Ⓣ(On the decomposition of definite functions in squares) also published in 1927.
- The path which led Artin to his reciprocity law began while he was still a student.
- Artin took the work of Takagi forward making several major steps.
- In 1923 in Über eine neue Art von L-Reihen Ⓣ(On a new type of L-series) Artin was able to obtain special cases of the results which were clearly forming in his mind and these special cases depended on the use of existing reciprocity laws.
- It was not Chebotaryov's result which was seen to be so important for Artin's theories, rather it was a method he used in his proof.
- With this idea as a basis Artin was able to reverse his 1923 approach.
- Instead of using the existing reciprocity laws, Artin proved his theorems based on the new approach which then yielded a new reciprocity law which contained all previous reciprocity laws.
- The theorems of Artin's 1927 paper have became central results in abelian class field theory.
- Similarly, Artin's Reciprocity Law opens the way to new applications and progress.The most striking application was given by Furtwängler's proof of the principal ideal theorem of class field theory, given one year after the publication of Artin's Reciprocity Law.
- Another important piece of work done by Artin during his first period in Hamburg was the theory of braids which he presented in 1925.
- Artin made a number of conjectures which have played a large role in the development of mathematics.
- In his seminal Ph.D. thesis Artin verified this in a number of cases numerically.
- Thus, this conjecture of Artin was the origin of a wide range of activities in what is now called arithmetic geometry.
- Second, there is Artin's conjecture on primitive roots.
- Given any integer g not 1 or -1, and g not a power of some other integer, then Artin conjectured that there are infinitely many prime numbers p such that g is a primitive root modulo p in the sense of Gauss.
- Artin made this conjecture to Hasse on 27 September 1927 (according to an entry in Hasse's diary), and since then many mathematicians have tried to prove it.
- Again, Artin's conjecture triggered a lot of interesting activities in number theory.
- Artin was not a Jew and was not affected by these laws.
- During his years in the United States Artin put his energies into teaching and supervising his Ph.D. students who themselves went on to make major contributions.
- In 1944 he did fundamental work on rings with the minimum condition on right ideals, now called Artinian rings.
- Among Artin's main books are Galois theory (1942), Rings with minimum condition (1948) written jointly was C J Nesbitt and R M Thrall, Geometric algebra (1957) and Class field theory (1961) written with J T Tate.
- In 1958 Artin returned to Germany, being appointed again to the University of Hamburg which he had left in such unhappy circumstances over 20 years before.
- Artin had many interests outside mathematics, however, having a love of chemistry, astronomy and biology.
- Artin was honoured by the award of the American Mathematical Society's Cole Prize in number theory.

Born 3 March 1898, Vienna, Austria. Died 20 December 1962, Hamburg, Germany.

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Algebra, Group Theory, Origin Austria

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive