D. K. Faddeev and St. Petersburg algebraic school

Abstract

A review of scientific achievements of D.K. Faddeev. DOI: 10.3103/S1063454108010019 TO THE 100th ANNIVERSARY OF BIRTHDAY OF DMITRII KONSTANTINOVICH FADDEEV On June 30, 2007, we celebrated the 100th anniversary of birthday of an outstanding mathematician, Corresponding Member of the USSR Academy of Sciences, Dmitrii Konstantinovich Faddeev (1907– 1989). Dmitrii Kostantinovich is the founder of the Leningrad–St. Petersburg algebraic school, which has gained world-wide recognition. Disciples of that school obtained a lot of scientific results of highest quality. He was born into the family of a St. Petersburg engineer in a small town of Yukhnov in Smolensk region (now Kaluga region). At the age of sixteen, he entered the mathematical department of Petrograd university and since that time to the end of his life (with a short interval in the late 1920s) his multifarious creative activity was closely related to our university, of which he became a professor in 1933. For a long period of time, Dmitrii Konstantinovich headed the chair of Higher algebra and number theory; during several years, he was dean of the department of mathematics and mechanics. In 1932, he started working at the Steklov mathematical institute of the USSR Academy of Sciences, which was at that time known as the Steklov physico-mathematical institute. From the moment of foundation of the Leningrad branch of the institute (LOMI, now POMI RAS) in 1940 to the end of his life, D.K. Faddeev was one of the most prominent researches there. For many years he headed the laboratory of algebraic methods, which became a generally recognized center of algebraic studies. The area of scientific interests of D.K. Faddeev was extremely wide. First of all, we remember him as one of the most prominent algebraists of that time. His investigations covered almost all subfields of algebra. His first results relate to the theory of Diophantine equations, which became a subject of his study under the influence of his scientific advisor B.N. Delone. These results are really impressive: he has essentially extended the class of equations of degree three and four which can be completely solved. As is known, solutions to binary equations of degree three form a group under the natural multiplication, and the Mordell– Weil classical theorem states that this group is finitely generated. However, the rank of this group is hard to calculate. In a number of cases, estimates for this rank obtained by D.K. Faddeev allow one to find all solutions to the equation. For instance, equations x 3 + y 3 = A are completely solved for all A ≤ 50. To illustrate the significance of this result, we note that, before Faddeev’s works, it was known only that, for some values of A , this equation possesses only trivial solutions. There is another elegant result concerning the equation x 4 + Ay 4 = ± 1: D.K. Faddeed demonstrated that, for any integer A , this equation has at most one nontrivial solution; moreover, this solution corresponds to the principal unit of a certain purely imaginary field of algebraic numbers of degree four and exists only when the principal unit is trinomial. These results were published in the remarkable monograph “The Theory of Irrationalities of the Third Degree” written by Dmitrii Konstantinovich in co-authorship with B.N. Delone. In later years, D.K. Faddeev continued investigating Diophantine equations. In a series of works he proves that the group of classes of divisors of degree zero for the Fermat curve of degree 4, 5, and 7 is finite. This yields very powerful results concerning the equations themselves. For instance, there exists only a finite number of quadratic fields where equation x 4 + y 4 = 1 possesses a nontrivial solution and for each of these fields, the number of solutions is finite. Results of this kind had never been obtained before. Of course, by now, many classical problems of the theory of Diophantine equations have been solved and in our days these results are viewed from another standpoint, but at that time, they were really outstanding. Another sphere of interests of D.K. Faddeev in the early years of his scientific activity was Galois theory. He was particularly interested in the so-called inverse problem of Galois theory (which still remains not completely solved). This problem consists in constructing an extension of a given base field with a pre-