F-K Schmidt was strongly influenced by two mathematicians who were just a couple of years older than him, namely Helmut Hasse and Wolfgang Krull. The influence was both through their mathematics and through personal contacts. Schmidt notes in the preface to his doctoral thesis that the topic was suggested by Krull who had been, in practice, Schmidt's thesis advisor. Krull had obtained his doctorate from Freiburg in 1922 and was a privatdocent during the time F-K Schmidt was undertaking research. Hasse, who moved from Kiel to Halle in 1925, corresponded with F-K Schmidt by letter - 150 letters from Schmidt to Hasse, which begin in 1926, were found in his papers at the time of his death. The influence of Krull and Hasse on F-K Schmidt can be seen in the outstanding work that he began to produce on the algebraic theory of function fields in one variable, on algebraic number theory and on valuation theory. In the autumn of 1926, he was appointed as an assistant to Otto Haupt at Erlangen and he habilitated there in the following year. His habilitation thesis, Abelian fields in the domain of higher congruences consisted of two parts: Part I: Analytic number theory in fields of characteristic p; and Part II: Class field theory for an algebraic function field of one variable with finite base field. He published the two parts of the thesis as separate papers, both appearing in 1931. Peter Roquette writes :-
... the first part of F K Schmidt's Habilitationsschrift, i.e., the paper [Analytic number theory in fields of characteristic p], became fundamental for the development of the theory of function fields. ... The first main achievement of F K Schmidt is the discovery that the classical theorem of Riemann-Roch on compact Riemann surfaces can be transferred to function fields with finite base field. Actually, his proof of the Riemann-Roch theorem works for arbitrary perfect base fields, not necessarily finite. ... F K Schmidt's second main achievement in [Analytic number theory in fields of characteristic p] is the discovery that over a finite base field, the Riemann-Roch theorem is intimately connected with the properties of his zeta function.We should note at this point that, later, André Weil made three conjectures that became known as the "Weil Conjectures". The third of these conjectures was a generalisation of the Riemann hypothesis to zeta functions. The Weil Conjectures were solved by Pierre Deligne in 1974 and he was awarded a Fields Medal for this achievement at the International Congress of Mathematicians in Helsinki in 1978. Jean Dieudonné, writing in  about the solution of the Weil Conjectures, states that F-K Schmidt was one of the main contributors of essential ideas to the ultimate solution.
In 1927 F-K Schmidt married Josefine Baumann, who had been a mathematics student in the same class as Schmidt at Freiburg. They had four children, Christa, Ursula, Elmar and Hanno. Christa and Elmar both went on to study mathematics, Elmar being awarded a doctorate in 1969 for his thesis Funktionentheoretische Charakterisierung der Topologie im Raume der gemässigten Distributionen .
In  Peter Roquette gives an interesting account (based on F-K Schmidt's letters to Hasse) of how F-K Schmidt solved an outstanding question in valuation theory:-
In the first week of February, 1930, F K Schmidt gave a colloquium talk at the University of Halle at the invitation of Hasse. The subject of the talk was the axiomatic description of those complete valued fields which admit local class field theory. On the occasion of this visit Hasse had posed (among other questions) the [following] question to F K Schmidt:Immediately after his return to Erlangen. F K Schmidt sent a letter to Hasse and said that now he can go farther than they were able to in Halle; he announces a more detailed letter soon. And already on 14 February 1930 he wrote:
Is it possible that a field K is complete with respect to two of its nonequivalent valuations?
Concerning the questions which we discussed in Halle, I would like to write you about two items. The first item is the Theorem: If the field K is complete with respect to a discrete valuation D then D is uniquely determined in two ways. ...F K Schmidt explains, what is of course evident, that "unique" in this context means "unique up to equivalence" of valuations. He continues to give a proof of this theorem. At the end of the letter he says:
Now I would like to think about whether a field can be complete with respect to two different valuations, which then of course cannot be discrete ...Two weeks later, in a letter of 29 February 1930, F K Schmidt tells Hasse that now he has completely solved the problem, including the case of non-discrete valuations, and he attaches to his letter a corresponding manuscript. That manuscript is not preserved but we may assume that it coincides more or less with F K Schmidt's paper in the 'Mathematische Annalen'; Mehrfach perfekte Körper (1933). At the annual meeting of the Deutsche Mathematiker Vereinigung in September 1930 in Königsberg, F K Schmidt gave a talk about [this] question and his solution. Hasse attended that talk and made notes about it; the content of his notes reflect the results of F K Schmidt's  paper.
The Nazis came to power in 1933 and this changed to face of German mathematics as they introduced their anti-Semitic legislation. Emmy Noether was dismissed from her position in Göttingen and, on 29 July 1933, Hermann Weyl began negotiating with F-K Schmidt trying to arrange for him to hold a visiting position at Göttingen and give the algebra courses that Emmy Noether had been giving. Weyl, however, resigned his position at Göttingen before the deal was finalised but his replacement, Franz Rellich, continued the discussion with Schmidt who accepted the visiting position. Also in 1933, F-K Schmidt succeeded Richard Courant as editor of Springer-Verlag's famous "Yellow Series" of mathematical monographs when Courant was dismissed because he was Jewish. F-K Schmidt was a Roman Catholic, and not Jewish, but he was quickly out of favour with the Nazis when he refused to remove Richard Courant's name from the title page of the Springer series. He also maintained contacts with his Jewish colleagues which displeased the Nazis. His time at Göttingen was difficult with continued arguments about which staff were "politically correct" and he was, for a time, temporary director of the Institute before Hasse took up the position.
In October 1934 F-K Schmidt was called to Jena as ordinary professor of mathematics and director of the mathematical institute. He continued to produce outstanding mathematical work and his paper Zur arithmetischen Theorie der algebraischen Funktionen I. Beweis der Riemann-Rochschen Satzes für algebraische Funktionen mit beliebigem Konstantenkörper (1936) contains an important proof of the Riemann-Roch theorem. Details are given by Ernst Kunz in . An interesting episode, which took place in 1939, is related to the setting up of Mathematical Reviews. It is told by Volker Remmert in  (see also ):-
Süss tried to put direct pressure on the two publishers de Gruyter and Springer in order to induce them to fuse. But the Springer Verlag had its own plans, namely to discuss the situation with the Americans first and, if possible, to cooperate with the 'Mathematical Reviews'. Ferdinand Springer wanted to send his main mathematical adviser, Friedrich Karl Schmidt, to the United States as a spokesman for his interests in mathematical reviewing. When Süss learned about this, he pressed Dames in the Ministry of Education and Research to refuse Schmidt permission to travel ... Süss was strongly opposed to Schmidt's journey. Schmidt, he had said, still had close ties to Jewish emigrants, and he suggested that he, Süss himself, should go to the United States instead. ... Süss explicitly characterised the 'Zentralblatt' as a foundation of "a group of Jewish mathematicians and their friends" and suggested that Schmidt's travel permission should be revoked and immediately because Schmidt intended to leave for the States the following week. Two days later, on April 29, Süss phoned Kummer in Berlin to inquire, how things stood. When Kummer informed him that Schmidt had already left, Süss told him that to his knowledge Schmidt was only on his way to Bremen to board the ship, which was due from America on May 1 or 2. Kummer did not take up the implication that Schmidt could still be stopped, but explained that his superior in the Ministry of Education and Research had definitely decided to let Schmidt go as he was not only to discuss 'Zentralblatt'-matters, but also to evaluate the atmosphere among American mathematicians and, if possible, to change their minds. It seems that Süss lost his temper and told Kummer that this was an unsuitable job for Schmidt and that the ministry would have done better to get the opinion of somebody who knew what was going on. The decision to send Schmidt he said was asking for trouble. Schmidt went on his mission to the United States, but nonetheless the 'Mathematical Reviews' came into being in 1939.In fact, when he returned from the United States, F-K Schmidt wrote a report for the Ministry of Education and Research saying that the founding of Mathematical Reviews could still be prevented but fusing de Gruyter and Springer would anger the Americans pushing them into founding Mathematical Reviews. The Ministry accepted Schmidt's report keeping de Gruyter and Springer separate but, in May 1939, the American Mathematical Society moved to found Mathematical Reviews which, when the Germans learnt of it, caused Schmidt problems. Soon after the start of World War II, in September 1939, Süss accused Schmidt of giving a false report after his return from the United States.
Stanford Segal writes in  about F-K Schmidt's difficult time in Jena:-
His continuing conflict with the Nazi authorities in Jena led to his gradually being sidelined as institute director, and even to his removal from the university's examination committee. By late 1941, the situation was such that Schmidt gave up his professorship at Jena and, with the help of applied mathematician Richard Grammel, found a position doing research on glider flight at Ainringen [near Bad Reichenhall], where he remained until the end of the war. [He was] known for his politeness, as well as his ironic humour ...After the end of World War II, in November 1945, F-K Schmidt returned to his professorship at the University of Jena. In July 1946 he wrote to Ferdinand Springer and asked him to write a letter of recommendation for him to the University of Münster where he was being considered for a professorship :-
He suggested that Springer mention that he had been known to cooperate with Jewish mathematicians as late as the end of 1938, and that his journey to the United States in May 1939 had been heavily opposed. In his letter of recommendation to the University of Münster, Ferdinand Springer ... followed Schmidt's outline ...F-K Schmidt took up the chair at the University of Münster from October 1946. He was offered a chair by the Humboldt University in Berlin in 1947 which he turned down (he spent a semester as a visiting professor there in 1947) and he also turned down the offer of a chair from the University of Erlangen in 1948. However, he was tempted by an offer from the University of Heidelberg which he took up in the summer of 1952. He continued to teach at Heidelberg until 1966 when he was made Professor Emeritus. He married Dr Anna Breassu in 1968.
As a lecturer F-K Schmidt was held in high regard. Dieter Puppe writes :-
Anyone who has attended a course by F-K Schmidt will not forget the enjoyment of his clear and sophisticated manner of performance down to the smallest detail. His lectures were always well attended, even if he delivered them at his preferred time of very early in the morning, and even after his retirement. ...We have indicated areas on which F-K Schmidt undertook research throughout this biography. However, for completeness we note that the authors of  put his work into seven areas: 1. Algebraic function fields of one variable; 2. Zeta functions; 3. Class field theory; 4. Valuation theory; 5. Galois theory and algebraic equations; 6. General field and ring theory; and 7. Derivations and differentials. Among the honours that Schmidt received for this work we mention his election to the Heidelberg Academy of Sciences in 1954 and the honorary degree that he was awarded by the Free University of Berlin in 1968.
Let us end this biography with a tribute from Dieter Puppe :-
For all his merits as a scholar and teacher, there is something else that gave rise to my admiration for him. I did not study with him, and my mathematical work has only a few points of contact with his area. But I got to know his wise decisions, his energy when doing the right thing, his sense of responsibility to the people entrusted to him, and the heartfelt courtesy with which he treated everyone. He was a role model, his counsel will be sorely missed, and the gap he leaves will be felt for painfully long.
Article by: J J O'Connor and E F Robertson