After leaving Hamilton Collegiate Institute, Fields entered the University of Toronto in 1880 to study mathematics. His undergraduate years were not too easy, for having already lost his father, his mother died while he was in the middle of his university course. Despite this he had an outstanding undergraduate record and he received his B.A. in mathematics from the University of Toronto in 1884, winning the gold medal for mathematics. He then went to the United States to study for his Ph.D. at Johns Hopkins University. After the award of the degree in 1887 for his thesis Symbolic Finite Solutions, and Solutions by Definite Integrals of the Equation (dn/dxn)y - (xm)y = 0, he remained teaching at Johns Hopkins for a further two years.
Fields was appointed in 1889 Professor of Mathematics at Allegheny College, one of the oldest colleges in Pennsylvania, but resigned after three years so that he might further his mathematical researches by studying in Europe. From 1892 to 1900 Fields studied in Paris, Göttingen and Berlin with Fuchs, Frobenius, Hensel, Schwarz, Weierstrass, and Planck. He was awarded the gold medal for mathematics at Berlin. He also met and formed a life-long friendship with Mittag-Leffler. This period was clearly important for his future development as a research mathematician. Synge writes in :-
This long period of study, which exercised a decisive influence on his life and outlook, was rendered possible by a modest private income, combined with simple living and abstemious habits.In 1902 Fields was appointed to the position of lecturer at the University of Toronto where he remained until his death, although he frequently visited Europe where he was acquainted with many national leaders. For example he attended a dinner party given by the King of Sweden in 1912. He became an Associate Professor in 1905 before being promoted top full Professor in 1914. In 1923 he was promoted to Research Professor at the University of Toronto.
His main research topic was on algebraic functions. He wrote an important book which was published in 1906. The main purpose of the book was to present the Riemann-Roch Theorem, the Weierstrass Gap Theorem and the related Hurwitz Theorem, and theorems of Brill and Max Noether. A colleague and former pupil of Fields gives this summary of his contributions in :-
The work of Professor Fields on algebraic functions must be regarded as the development and organisation of ideas on the subject which he happened to have at the turn of the century. These had very little reference to prevailing methods. He made it his life work, first to show that they could be used to create a satisfactory theory and then to give to the structure thus secured both elegance and generality. His treatment has the great merit of being completely algebraic in character and of meeting every difficulty without an appeal to geometric intuition. The machinery, which he had to invent for the purpose, is simple, and its parts are beautifully coordinated.It was, however, rather as an organiser of mathematics that Fields excelled. The series of International Congresses of Mathematicians began in Zürich in 1897 but no congress was held during World War I (1914-18). The International Mathematical Union was set up in 1920 at the first post war Congress in Strasbourg to run future congresses but, in the aftermath of war, Germany, Austria-Hungary, Bulgaria, and Turkey were excluded from the Union. This was unfortunate and feelings ran high on the issue. Many felt that mathematics should not be subjected to political pressures but should welcome equally mathematicians of all nations. Leading advocates of that view were Hardy and Mittag-Leffler. Others held equally sincere views that the Union's exclusion rules were right.
In 1922, following the collapse of a bid from New York to hold the 1924 Congress, Fields made a bid to hold it in Toronto under the auspices of the International Mathematical Union, and although this meant that the congress could not be truly international because of the exclusion rules, it did prevent a break-up of the congress. He was a skilled politician and was able to persuade many opponents of the Union to attend the Toronto Congress and have it succeed. This was not easily achieved and Fields spend several months in Europe working relentlessly to make the Congress a success. He also had to get financial support so that Europeans could be helped with their travel costs to North America. So successful was he in getting financial support that he had money left at the end of the Congress. It gave him the opportunity to come up with a wonderful idea.
Fields is best remembered for conceiving the idea of, and for providing funds for, an international medal for mathematical distinction. The original proposals were put on 24 February 1931 to the committee who had run the 1924 Congress, and money left over from the finances were to be used. Fields had everything in place to travel to the September 1932 Congress in Zürich to put forward his proposal for Medals. He had already done the ground work and by January 1932 he had support for awarding Medals from the leading mathematical societies in France, Germany, Italy, Switzerland and the United States.
You can see Fields's Letter setting out his proposals at THIS LINK.
However, his health began to fail in May of 1932 when he suffered heart problems. A few days before his death he drew up a will, with Synge at his bedside, including an amount of $47,000 to be added to the funds for the medals. He did not live to attend the Congress but his plans were still put forward. Adopted at the International Congress of Mathematicians at Zürich in 1932, the first Fields Medals were awarded at the Oslo Congress of 1936. Notice that they were named "Fields Medals" despite his wish that they should not bear anyone's name.
Fields Medals were to be awarded to two mathematicians under 40 years of age every four years at the International Congress of Mathematicians. These conditions were set down to recognise Fields's wish that the awards recognise both work completed and point to the potential for future achievement. Notice that the 40 age limit was not explicitly due to Fields. The first Fields Medals were awarded to Lars Ahlfors and Jesse Douglas in 1936. No awards were made during World War II, then beginning from 1950 the Medals have been awarded every four years. In 1966 it was decided that no fewer than two Medals would be awarded at each Congress, and no more than four.
A prize of 15,000 Canadian dollars is awarded with each Fields Medal, which is made of gold, and shows the head of Archimedes. A quotation attributed to Archimedes is inscribed:-
Transire suum pectus mundoque potiri.The inscription:-
(Rise above oneself and grasp the world.)
Congregati ex toto orbe mathematici ob scripta insignia tribuereappears on the reverse.
(The mathematicians assembled here from all over the world pay tribute for outstanding work)
Fields received several important honours. He was elected a fellow of the Royal Society of Canada in 1907 and, in 1913, he was elected a fellow of the Royal Society of London. In 1924 the International Congress of Mathematicians was held at Toronto and Fields was honoured by being President of the Congress. He was vice-president of the next International Congress of Mathematicians at Bologna in 1928 (at which the excluded nations were readmitted). He also held the position of President of the Royal Canadian Institute from 1919 to 1925. He was president of the International Mathematical Union, vice-president of the British Association for the Advancement of Science, and vice-president of the American Association for the Advancement of Science all in the year 1924. He was also elected a member of the Russian Academy of Sciences and of the Institute of Coimbra. An interesting aside is that the Italian Government wanted to honour Fields with the title "Commander of the Crown of Italy", but the Canadian Government had a law forbidding Canadian citizens from holding titles, so Fields had to refuse the Italian government's offer.
Let us end this biography by giving a few details of Fields's life outside the world of mathematics. He never married. As a young man he loved sport, playing baseball, football and hockey. He also loved walking. Music was a great joy to him and he played the violin, and loved dancing :-
In his habits of life he was abstemious, avoiding tea, coffee, alcohol, and condiments, and he did not smoke ... he possessed a good sense of humour ...
Article by: J J O'Connor and E F Robertson