Artin's childhood was not a particularly happy one and he recounted later in his life how he had felt lonely. He did not find himself attracted to mathematics at a very young age, contrary to what seems to happen to most mathematicians, and up to the age of sixteen the subject meant no more to him than any of his other school subjects and less than some. Rather surprisingly up to sixteen he did not show any particular talent for the subject; at least this was his own view of his schooldays when he spoke of them later in his life. The school subject which he did show talent for, and which he was most attracted towards, was chemistry. He spent a happy school year in France, the happiest of his schooldays, then his interests moved towards mathematics during his final two years at school.
By the time he took his school leaving examinations in 1916 in Reichenberg, Europe had already suffered two years of World War I. However Artin did begin his university career, enrolling at the University of Vienna. After one semester, however, he was drafted into the Austrian army and he served with this army until the end of the War. Then, in January 1919, he entered the University of Leipzig where he continued his mathematics studies with Herglotz. Academic success came quickly and in 1921 he was awarded his doctorate. His thesis concerned applying the methods of the theory of quadratic number fields to quadratic extensions of a field of rational functions of one variable taken over a finite prime field of constants. After receiving his doctorate he attended the University of Göttingen for one year (1921-22). He went to the University of Hamburg as an assistant in October 1922 for the start of winter semester of session 1922-23. In 1923 he had his Habilitation and accordingly became Privatdozent at Hamburg.
At Hamburg Artin lectured on a wide variety of topics including mathematics, mechanics and relativity. He was promoted to extraordinary professor there in 1925, then he became an ordinary professor in the following year. These were particularly productive years for Artin's research. Brauer wrote in :-
The ten year period 1921-1931 of Artin's life [saw] an activity not often equalled in the life of a mathematician.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. He had the distinction of solving, in 1927, one of the 23 famous problems posed by Hilbert in 1900. Also in 1927 he gave a general law of reciprocity which included all previously known laws of reciprocity which had been discovered from the time that Gauss produced his first law.
Field theory had been created by Steinitz in 1910. 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. The problem which he solved was whether, given an algebraically closed field O, there exist subfields K, properly contained in O, with O an algebraic extension of finite degree of its subfield K. In his 1924 attack on this problem Artin restricted himself to considering only fields which were an algebraic closure of the field of rationals. However, two years later in 1926 he realised that his arguments actually proved more than he had originally thought, and he was able to solve the problem for any algebraically closed field of characteristic 0. By this stage he had proved, using very clever arguments with Galois theory and Cauchy's theorem on subgroups of prime order, that O had to be an extension of K of degree 2 and that the subfield K had to have the property that -1 could not be expressed as a sum of squares. 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 p. In fact, in the case of prime characteristic, they proved that the field O cannot be a finite extension of a proper subfield K.
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. They also defined real-closed fields to be those that were formally real yet every algebraic extension of them failed to be formally real. Artin himself proved that when O is the field of algebraic numbers, the subfield K of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of the field O. 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 Funcktionen in Quadrate Ⓣ also published in 1927. It is also worth noting that the theory of real-closed fields directly influenced Abraham Robinson in his contributions to model theory, particularly for the concepts of model completeness and model completion, see for example .
The path which led Artin to his reciprocity law began while he was still a student. In 1920 Takagi published his fundamental paper on class field theory in which he built the theory around a remarkable fact which he had discovered, namely that the set of class fields, as defined by Heinrich Weber, over a fixed ground field k is identical to the set of abelian extension fields over k. Artin took the work of Takagi forward making several major steps. He defined a new type of L-series, which generalised Dirichlet's L-series, yet was quite different in nature. In 1923 in Über eine neue Art von L-Reihen Ⓣ 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. However, in 1927 he published his masterpiece on the subject Beweis des allgemeinen Reziprozitätsgesetzes Ⓣ where now he had developed the results differently.
The new idea originated in work which Nikolai Chebotaryov published in 1924 where he had proved a conjecture made by Frobenius about the density of the set of prime ideals of a normal extension field. 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. Roquette writes :-
In my opinion, the main importance of Artin's Reciprocity Law is that it opens a new viewpoint on those classical laws, formulating it as an isomorphism theorem. The situation is similar to that with Galois Theory which, today, is formulated in the framework of abstract algebra, and in this form opens new applications and generalizations. 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. He again showed his originality by introducing this new area of research which today is being studied by an increasing number of mathematicians working in group theory, semigroup theory, and topology.
Artin made a number of conjectures which have played a large role in the development of mathematics. Two of these, mentioned by Roquette in , have wide interest, namely:-
First, the analogue of the Riemann conjecture for the zeta function of a curve over finite fields. In his seminal Ph.D. thesis Artin verified this in a number of cases numerically. In 1933 Hasse succeeded in proving this for elliptic curves, and in 1942 Weil for arbitrary curves. Later, as is well known, Deligne generalized this for arbitrary varieties. Thus, this conjecture of Artin was the origin of a wide range of activities in what is now called arithmetic geometry.Artin married one of his students, Natalie Jasny, in 1929 :-
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. More precisely: the set of those prime numbers has positive density, which can be written down and computed explicitly. 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. Hooley has proved it under the condition that a strong form of Riemann's hypothesis (for number fields) is valid. There are very interesting unconditional results, proved by Heath-Brown and others. Again, Artin's conjecture triggered a lot of interesting activities in number theory.
His family now occupied a central position in his life. When his children were growing up, he took a most active part in all phases of their education. He spent hours with them every day, and it was of foremost importance to him to instil in them his own personal and cultural standards.On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Artin was not a Jew and was not affected by these laws. However his wife was a Jew so when the 'New Official's Law' was passed in 1937 those related to Jews by marriage were affected. Artin :-
... with his feeling for individual freedom, his sense of justice, his abhorrence of physical violence ...had no real alternative but to leave Germany. In 1937 he emigrated to the United States and taught at various universities there. He was at Notre Dame for the academic year 1937-38, he spent eight years at Bloomington at Indiana University from 1938 to 1946, and then he was twelve years at Princeton from 1946 to 1958.
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. He published relatively few papers, but he wrote a number of extremely important texts which have become classics. In 1944 he did fundamental work on rings with the minimum condition on right ideals, now called Artinian rings. He presented new insight into semi-simple algebras over the rationals. In 1955 he produced two important papers on finite simple groups, proving that the only coincidences in orders of the known (in 1955) finite simple groups were those given by Dickson in his Linear groups. This important piece of work is one of a number of results leading to the intense interest in finite simple groups which eventually led to their classification.
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.
Perhaps his views on teaching and writing texts are best illustrated by a quotation from a review he wrote in 1953:-
We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception should be instantaneous. We have all experienced on some rare occasion the feeling of elation in realising that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications.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. He made the decision to return to Germany in 1956, for in that year he took his first ever sabbatical leave which he spent in Germany. It was his first visit to that country since he left it in the grip of the Nazis in 1937. During his sabbatical leave he revisited universities which had a special place in his mathematical achievements. He taught for a term in Göttingen and then returned to Hamburg where he also taught for a term. In 1958 Artin returned to Hamburg and, in a moving passage in , Brauer describes walking through the streets of Hamburg with Artin in November 1958:-
We took a long walk one afternoon talking of old times. It was one of those misty, melancholy, and rather miserable days which all northern harbour cities know so well in late fall. We wandered endlessly through the streets searching, I did not know for what, until I realised, it was a Hamburg which no longer existed and times which were gone for ever. Before Artin's eyes, I believe, there must have been the picture of the young Artin who had walked through the same streets thirty years before, full of life and strength.Artin had many interests outside mathematics, however, having a love of chemistry, astronomy and biology. He also loved music and was an accomplished musician playing the flute, harpsichord and clavichord. Roquette writes :-
I remember in Hamburg when he once told me of a conference on electronic music which he had attended.An amateur astronomer, he even built his own telescope as a hobby.
Artin was honoured by the award of the American Mathematical Society's Cole Prize in number theory. In  his influence is described as follows:-
Artin's scientific achievements are only partially set forth in his papers and textbooks and in the drafts of his lectures, which often contain new insights. They are also to be seen in his influence on many mathematicians of his period, especially his Ph.D. candidates (eleven in Hamburg, two in Bloomington, eighteen in Princeton).
Article by: J J O'Connor and E F Robertson