After graduating from the Gymnasium, Rychlik entered the Charles-Ferdinand University in Prague in October 1904. He spent the session 1907-08 in Paris. At the Faculty of Science he attended lectures by a number of outstanding mathematicians such as Jacques Hadamard, Émile Picard, Gaston Darboux, Édouard Goursat, Louis Raffy, and Paul Painlavé. He also attended Georges Humbert's lectures on number theory at the Collège de France. After he returned to Prague he completed the examinations to qualify him to teach in secondary schools in December 1908 and in March of the following year he was awarded his doctorate for his thesis on substitution groups (permutation groups in today's terminology). Already before his doctorate had been conferred, Rychlik was appointed as an assistant at the Charles University in Prague although for the first year he was unpaid. He began to receive payment from December 1909 and then began to submit papers to be considered as his habilitation thesis. The first of these was A Contribution to the theory of forms (submitted November 1910), followed by A Contribution to the theory of forms II (submitted June 1911). After giving an address The Evolution of the Concept of Divisibility he was appointed as a dozent in January 1912. In this role Rychlik gave lectures on the Theory of Algebraic Fields and on the Theory of Algebraic Functions.
In 1913 Rychlik's younger brother Vilém died at the age of 26. Vilém had, like his older brother, studied mathematics and physics at the Charles University. After the award of his doctorate he had been appointed as an assistant in mathematics at the Czech Technical University in Prague and had a very promising career in front of him when he caught a cold and died three days later. Perhaps the fact that he smoked 40 cigarettes a day was a factor.
For Rychlik, having a position at the Charles University of Prague was a good thing in terms of status but it did not provide a sufficient income for him to live. So in addition he had to take on other employment which he did from July 1913 at the Czech Technical University in Prague. When World War I broke out, Frantisek Velisek who was a professor at the Technical University, enlisted in the army; he was killed in the war. Rychlik took over his lecture courses at the Technical University. He married in 1918 but it was two years later, in November 1920, that he was appointed as an extraordinary professor at the Technical University and for the first time received a good salary.
Rychlik's contributions cover a number of different areas. He did excellent work on algebra and number theory, for example he generalised Hensel's ideas on g-adic numbers in 1914, later approaching them via sequences and limits unlike the 'generalised decimal expansion' approach of Hensel. He introduced the theory of pseudo-valuation in a paper in 1916, twenty years before the corresponding definition by Mahler, who is usually considered the founder of the theory. In fact it is easy to see why Rychlik did not receive as much credit for this work as one might expect for many of his papers were written in Czech and therefore not read outside his country. Rychlik is best known, however, for his contributions to the history of mathematics, particularly his work on Bernard Bolzano.
On 5 March 1924 the Czech Academy of Sciences set up the Bolzano Committee to edit and publish Bolzano's manuscripts. Rychlik was one of eight members of this Committee and it was he who edited and provided notes for the first of the Committee's publications 'Bernard Bolzano, Spisy Bernarda Bolzano - Bernard Bolzano's Schriften Vol. 1, Functionenlehre, Edited and with notes by K Rychlik (Královská Ceská Spolecnost Nauk, Prague, 1930).' This publication comprised an edited version of Functionenlehre Ⓣ which was written by Bolzano before 1834. It contains the famous Bolzano function which is continuous on an interval [a, b] but not monotonic on any subinterval. Given two points in the interval at which no derivative exists, Bolzano showed there was an point lying between them at which no derivative exists. Rychlik also edited and provided notes for the second of the Committee's publications 'Bernard Bolzano, Spisy Bernarda Bolzano - Bernard Bolzano's Schriften Vol. 2, Zahlentheorie, Edited and with notes by K Rychlik (Královská Ceská Spolecnost Nauk, Prague, 1931).' It contains part of Bolzano's manuscript entitled Zahlentheorie Ⓣ , namely the part in which he presented the integers and their elementary properties.
When World War II broke out in 1939 all the Czech universities were closed down. By the time that the war ended in 1945 Rychlik was sixty years old and at that stage he thought he should retire; he formally retired from his university positions in 1948. However he began to undertake deep research in the history of mathematics, and in particular he worked again on Bolzano. Other works by Rychlik on Bolzano from this later period of his research career include Theory of real numbers in the manuscripts left by Bolzano (Czech) (1956), Theorie der reellen Zahlen im Bolzano's handschriftlichen Nachlasse Ⓣ (1957), Betrachtungen aus der Logik im Bolzano's handschriftlichen Nachlasse Ⓣ (Czech) (1958), Betrachtungen aus der Logik in Bolzanos handschriftlichem Nachlasse Ⓣ (1958), La théorie des nombres réels de Bolzano d'après ses manuscrits inédits Ⓣ (Russian) (1958), and Theorie der reellen Zahlen in Bolzanos handschriftlichem Nachlasse Ⓣ (1962). Other works by Rychlik on the history of mathematics include A Cauchy manuscript in the archives of the Czechoslovakian academy of sciences (Czech) (1957), Un manuscrit de Cauchy aux archives de l'académie tchécoslovaque des Sciences Ⓣ (1957), Cauchys Schrift "Mémoire sur la dispersion de la lumière" herausgegeben während seines Aufenthaltes in Prag durch die Königliche böhmische Gesellschaft der Wissenschaften Ⓣ (1958), and Berechnung der Grundzahl e der natürlichen Logarithmen Ⓣ (Czech) (1960). In this last mentioned paper Rychlik looked at the calculation of e to 225 decimal places carried out by the Czech mathematician Bohumir Tichánek in 1890 using a continued fraction method. The Cauchy manuscript which Rychlik discussed in the above mentioned papers was Memoire sur l'intégration des équations differentielles Ⓣ which was dated 'Prague 1835' in Cauchy's own hand. Because Cauchy left Prague in 1836, this manuscript was not printed, as he had intended, in the Proceedings of the Société Royale des Sciences de Bohême.
Hyksová writes in  about Rychlik's involvement with Societies and Academies:-
In 1904 Rychlik became a member of the Union of Czech Mathematicians and Physicists and until World War II he was also a member of its committee. Almost the whole of his life Rychlik lectured in the Union and his lectures were very closely related to his scientific research. He was also a member of the Royal Bohemian Society of Sciences (elected on 11 January 1922), the Czech Academy of Sciences and Arts (23 May 1924) and the Czechoslovak National Research Council under the Academy (19 May 1925).
Article by: J J O'Connor and E F Robertson