Cole returned to Harvard and wrote a thesis A Contribution to the Theory of the General Equation of the Sixth Degree which, as the title indicates, studied equations of degree 6. The topic had been suggested to him by Klein who was effectively his thesis supervisor. Cole presented this thesis to Harvard University, not to Göttingen University, for his Ph.D. which was awarded in 1886. While he completed writing his doctoral thesis, Cole had already begun to lecture at Harvard and he continued to lecture there until 1887. During the year 1887-88 Cole was a tutor at Harvard, then in the summer at the end of that academic year he married Martha Marie Streiff of Göttingen on 26 July; they had one daughter and three sons. In October 1888 he was appointed instructor at the University of Michigan, being promoted to assistant professor in the following year.
In 1895 Cole was appointed professor at Columbia University, a post which he held until his death. Soon after he arrived at Columbia University he was appointed as Secretary of the American Mathematical Society and he held this post from 1896 until 1920. This was not his only work for the American Mathematical Society for in 1897 he was appointed as Editor-in Chief of the Bulletin of the American Mathematical Society, holding this position until just before his death. He was vice president of the Society in 1921 but although he was offered the honour of being president of the American Mathematical Society, he turned it down.
His main research contributions are to number theory, in particular to prime numbers, and to group theory. In number theory he achieved the distinction of being the first to factor 267 - 1 and he did this using quadratic remainders. In fact
Another important contribution to the theory of groups made by Cole was his work in publishing his English translation of Netto's book on group theory. This appeared as The theory of substitutions and its applications to algebra (A A Register Publishing Company, 1892) and was undertaken with Netto's agreement. In fact Netto revised the German text of his book so that the English translation by Cole could incorporate some improvements. It was the first book on group theory in English and was important in stimulating interest in group theory in the English speaking world. It was reprinted in 1964.
As a lecturer Cole was described as "truly inspiring" by Osgood, who was not in general one to give praise easily. He brought the latest mathematical research topics to his lectures and was a major factor in improving mathematical education at Harvard. Osgood and M Bôcher were among his students.
He established the Frank Nelson Cole Prizes in algebra and number theory and today these are highly prestigious awards. The way this came about was that when Cole ended his spell of twenty-five years of secretarial and editorial duties for the American Mathematical Society his fellow members of the Society were so appreciative of his contributions that they collected money to give him to show their recognition. It was this money that he chose not to accept, but to return it to the Society to set up the prize.
Cole is described by David Smith in  as follows:-
As a man Cole was admired by all who penetrated a certain reserve that was natural to him, as an executive he was faithful to every duty, as a teacher he was lavish of the time that he would give to those who proved their worth, and as a friend he was loyal to the last. He loved to take long walks in the country studying trees and wild flowers.In fact Cole took his love of trees and wild flowers from his father, and in fact, as we mentioned near the beginning of this article, inherited his love of mathematics from the same source.
The biographical details we have given suggest that Cole's life was a rather straightforward one of devotion to mathematics and his family. However after he died a strange story came to light which has never been fully explained. Zund relates details in :-
In 1926 Cole announced his intention to retire from Columbia in September of that year. However in late May, following surgery for an infected tooth, he died of heart failure at his home in New York City. Subsequently, it became known that he had been estranged from his family since 1908, and for the last two years he had lived in the Bronx under the assumed name of Edward Mitchell and pretended to be an "ordinary bookkeeper". The reason's for Cole's unusual behaviour, which generated much comment and speculation, were only partially explained after his death.
Article by: J J O'Connor and E F Robertson