Kroll was born in the town of Frankenstein in the Prussian Province of Silesia and brought up in Breslau, the capital city.
Before the end of the term, he was awarded a four-year scholarship by the Prussian Academy of Sciences that gave him the means to further his academic career.
Building upon his success Kroll tried to secure a better position for himself and in 1913 finally obtained a chair at his alma mater in Breslau.
He was one of the first German scholars to be invited to lecture in the United Kingdom after World War I, and was awarded a visiting professorship at the Princeton University in 1930/31.
While not being a victim of the Nuremberg Laws himself, Kroll witnessed the removal of his colleagues from office and the persecution of his former pupils, some of whom he aided in finding work abroad.