Uspekhi Mat. Nauk 44:3 (1989), 187-193 Russian Math. Surveys 44:3 (1989), 223-231

(on his eightieth birthday)

Dmitrii Konstantinovich Faddeev, Corresponding Member of the Academy of Sciences of the USSR, celebrated his eightieth birthday in 1987.

The range of Faddeev's interests is unusually broadamong his more than 150 papers are some on the theory of functions, on computational methods, on probability theory, on problems of teaching mathematics at all levels. But, of course, Faddeev is known in the first place as one of the most outstanding algebraists of our time.

In his papers Faddeev touched on a wide range of algebraic problems. But there are two areas in which he began his research and to which he constantly returned. They are Diophantine equations and Galois theory.

Faddeev's very first results in Diophantine equations were remarkable. He was able to extend significantly the class of equations of the third and fourth degree that admit a complete solution. When he was studying, for example, the equation x3+y3 = A, Faddeev found estimates of the rank of the group of solutions that enabled him to solve the equation completely for all A <= 50. Until then it had been possible to prove only that there were non-trivial solutions for some A. For the equation x4+Ay4 = ±1 he proved that there is at most one non-trivial solution; this corresponds to the basic unit of a certain purely imaginary field of algebraic numbers of the fourth degree and exists only when the basic unit is trinomial.

Faddeev returned again and again to the study of Diophantine equations in the years that followed. In , ,  he studied the group of classes of divisors of zero degree for a Fermat curve of degrees 4, 5, 7 and he proved that it is finite: from that there followed very powerful results for the equations themselves. For example, there are only finitely many quadratic fields in which the equation x4+y4 = 1 has a non-trivial solution, and in each of these fields there are only finitely many solutions. Until then there had been no results of this kind.

Faddeev made a significant contribution to Galois theory. He was especially interested in the inverse problem: the construction of extensions (all or at least one) of a given field with a prescribed Galois group. In his earlier papers [I], , , , , he had solved this problem completely for small groups (subgroups of the group of permutations of three or four elements, metacyclic transitive groups of permutations of a prime number of elements, groups of quaternions and quaternion units). For this he used a beautiful geometric method: the unknown field is interpreted as a subset of a certain vector space, on which the action of a Galois group is described quite simply. Many of the results obtained here are presented in an elegant geometrical formulation. For example, extensions of the field of rational numbers with a quaternion Galois group prove to be closely connected with triples of mutually orthogonal vectors with rational components in three-dimensional Euclidean space.

Howver, similar arguments proved inadequate for further progress. In a remarkable paper of 1944 (, a joint paper with B.N. Delone) Faddeev began a systematic study of a problem which is much more general than the inverse problemthe problem of Galois embedding: to embed a given Galois extension in a wider field with a given Galois group and with a given epimorphism of this group onto the Galois group of the original extension. Here he stated a necessary decidability condition for the embedding problem the condition of compatibility, it consists in the additive decidability of the problem (it demands of the solution that it be a vector space on which a group of operators acts, properly compatible with the Galois group of the embedded field and no multiplication at all is required). It was shown there too that compatibility is sufficient for embeddability if the kernal of the embedding problem under consideration is a cyclic group of odd order. Moreover, Faddeev proved in the same paper that if the embedding problem with an Abelian kernel for number fields admits a solution that is a Galois algebra, then there is a field that solves it.

The compatibility condition can be formulated in various ways (some of them are stated in the same paper). It turned out to be so deep and so close to the sufficient condition for embedding that H. Hasse (who discovered it four years later for a special case) assumed it and tried to prove that is is a sufficient condition for embedding (at least in the case of an Abelian kernel). This error was made by other eminent mathematicians. However, as follows from the examples of Shafarevich and Faddeev, Hasse's conjecture is false. In Faddeev's example  the kernel of the embedding problem is the cyclic group of order 8. A supplementary necessary and sufficient condition for embedding in this case can be found in the paper , written jointly with R.A. Shmidt. For an arbitrary Abelian kernel a supplementary condition for embedding was found by A.V. Yakovlev, a pupil of Faddeev.

Faddeev's work on Galois theory proved to be very influential in the further development of this theory. In particular, it (together with Faddeev's results on cohomology of groups) played an essential role in Shafarevich's proof of the decidability of the inverse problem of Galois theory for soluble groups and number fields.

In 1947 Faddeev published a paper  in which he defined (at the same time as the American mathematicians Eilenberg and MacLane but independently of them) cohomology groups for groups. This created a powerful tool for use in different fields of algebra, especially in Galois theory: cohomologies were an ideal device for studying problems of embedding of fields, for constructing extensions with soluble groups, and for other problems connected with Galois theory and applications of it. In later papers Faddeev proved many important theorems in homological algebra: of particular interest to him was the link between the cohomologies of a group and a subgroup of it. Very important are his joint papers  and  with Borevich, in which not only was there given for the first time a systematic exposition of the theory of cohomology of groups, but new interesting results were obtained. In particular, a proof was given of the analogue of the Krull-Schmidt theorem for operator modules over complete local rings, minimal projective resolutions were constructed, and the existence of finitely many pairwise non-isomorphic indecomposable modules of p-adic representations of a non-cyclic p-group was proved.

In  and  the homological apparatus was effectively used in the study of simple algebras over the field of algebraic functions of one variable. A theory was constructed which in many ways is analogous to the theory of algebras over fields of algebraic numbers. In particular, in  the Brauer group of the field of rational functions over a field of characteristic 0 was calculated. This result was considerably ahead of its time, and was only proved again 17 years later by the American mathematicians Auslander and Brumer.

At the end of the 50's Faddeev turned to problems in the theory of integral representations. His numerous lectures and reports drew attention to this group of problems and contributed to the intense development of work on them. In a long article  the strategy was laid down for work in the field of integral representations. In this paper he put forward a multiplicative theory of lattices in algebras over the quotient field of a Dedekind ring; with the help of this theory it was possible to isolate six different levels for classifying these lattices. The most precise level was the classification up to a similitude; the crudest was the coincidence of coefficient rings. In some cases different levels of classification may coincide; for example, for commutative rings there are only three levels left (besides those already listed up to a local similitude). In this way the general problem of classification is divided into a series of steps, each of which requires an independent approach.

In a series of papers ( and jointly with Borevich , ) he applied this programme to specific rings. In  he studied cubic Z-rings, which were of special interest, since they are closely connected with earlier research by Faddeev and Delone into cubic irrationalities. In the other papers, written jointly with Borevich, they studied representations of orders with a cyclic index in the maximal order. They proved that every finitely generated torsion-free module over such an order decomposes into a direct sum of ideals of the order and gave a complete set of invariants that specify the module. Bass studied the same class of rings from a different angle;

using the results of  he showed that a cyclic index is not only a sufficient but also a necessary condition for the decomposability of a torsion- free module into a sum of ideals.

Faddeev's contribution to the theory of representations was not confined to his own results. Under his influence a powerful school was founded in the USSR which produced many outstanding results.

The papers , , ,  treat another aspect of the theory of representations. They contain a beautiful classification of complex representations for classical groups over finite fields in terms of Green's multiplication. For the full linear group the so-called simple representations are distinguished, which are not contained in products of representations of full linear groups of lower orders. The irreducible components of a power of a simple representation are called primary; they correspond to representations of the symmetric group of order equal to the degree. An arbitrary representation is uniquely represented in the form of a product of primaries. For the full affine group any irreducible representation is obtained by multiplying the so-called "pivotal" representations by arbitrary representations of the full linear group. The pivotal representation in every dimension is unique and is induced from the one-dimensional non-singular representation of the group of upper unitriangular matrices.

Faddeev is one of the leading experts in numerical methods of linear algebra. His work in this field was mostly done in collaboration with V.N. Faddeeva, and played an important role in establishing and developing this science. The survey lectures they gave at many all-union and international conferences, their survey articles on numerical methods of algebra (, , , ), their monograph  Vychislitel'nye metody lineinoi algebry (Numerical methods of linear algebra), published in 1960 and in revised form in 1963, played a great part both in the training of computational mathematicians and in the development of numerical methods in linear algebra. The monographs of Faddeev and Faddeeva won world-wide recognition, were translated into many languages, and are still relevant today. They contain a large number of methods and algorithms for solving algebraic problems, a profound analysis of the principles behind their construction, and a systemization of them. Many methods were further developed. Characteristic of Faddeev in his striking sense for what is new and promising. His talks, monographs, and articles are not only excellent guides through the labyrinth of published material, but they contain many ideas giving food for creative thought.

Most of Faddeev's own research is basically concerned with the stability of the numerical solution of systems of linear algebraic equations and estimation of the results of calculations. The stability of the solution under a variation of the input data (the conditioning of the system) and an investigation of its quantitative characteristics are considered in , , and . Here in particular we can clearly see the influence of Turing's condition number on the estimate of the error of the solution when the elements of a matrix are independent random numbers with a common small dispersion; in terms of the singular decomposition of the matrix the analysis of the system is carried out on the condition that makes it possible to clarify what information an ill-conditioned system carries within itself; here is posed and answered the problem of finding two diagonal matrices D1 and D2 yielding the minimum number of the Turing condition for the matrix D1AD2.

A new approach to estimating the complete error in the solution of a problem in finite calculation was developed in , , , . In particular, within the limits of the linearized theory of errors there is a proposal to describe the natural domain of solutions and to replace an estimate of its norm by an estimate of elliptic norms specially chosen for the given problem and closely approximating the natural norms. The construction of such elliptic norms is achieved by using the device of companion matrices. This method imitates the probability estimates of the linear probability theory of errors. The link between companion and correlation matrices is studied, and also the transition from norms in which the input data are given to elliptic norms, and an estimate of the loss of information occurring during this transition.

The papers  and  deal with the solution of linear systems of a general form, where from geometrical positions light is thrown on questions connected with the solution of systems with rectangular matrices. An algorithm is proposed for factorizing an arbitrary matrix into a left trapezoidal matrix with a certain prescribed ordering of elements and a matrix of orthonormalized columns. A link is established between the singular numbers of the matrix of the system to be solved and the diagonal elements of the left trapezoidal matrix. On the basis of the above mentioned factorization, numerically stable algorithms are proposed for solving systems with rectangular matrices, and also an algorithm for analysing ill-conditioned systems.

In  a new concept is suggested for estimating the quality of the results of the numerical solution of linear-algebraic systems depending on the quality of the specification of the input data. Three situations are considered:

scholarly (the input data are given precisely), regular (possible variations of the input data are far from critical), and irregular (variations of the input data are close to or coincide with the critical problem, which reduces to a qualitative variation). An analysis is given of the solution of the problem in each of these situations.

Of Faddeev's other papers we must mention [ 108] and . In [ 108] he investigates the properties of consistency and multiplicativity for norms in spaces of polynomial forms of vectors that belong to finite-dimensional spaces with vector norms. In  he proposes a method for embedding the algebra of matrices of lower order in the algebra of matrices of higher order, which could be used for the approximation of large problems by small ones, both for the solution of systems and in the spectral problem of matrices.

Problems in the teaching of mathematics in schools have long been one of the themes of Faddeev's scientific and teaching work. The book "Algebra" by Faddeev and I.S. Sominskii appeared as early as 1951 and was reissued in 1964, a book aimed at giving greater depth to the teaching of algebra courses in schools. Faddeev was one of the founders and organizers of the Physics and Mathematics Boarding School at Leningrad State University.

A distinctive feature of Faddeev's approach to the teaching of mathematics in schools is his clearly expressed and logically developed chain of ideas in the courses he devised on algebra and basic analysis. The essence of his ideas is expressed in his own words: "The elementary part of mathematics is the simplest of the sciences, since its object is a study of the crudest aspects of reality". Faddeev then goes on to subtle mathematical ideas of continuity, of limit, of smoothness of a function, of a derivative, by way of a practical, intelligent and interesting use of the concept of a number as the result of measurement or of a rationally and appropriately executed calculation. Faddeev gives priority to "vivid contemplation", to awareness, to direct perception of the properties of mathematical objects, which thus

reveal the content and basic ideas of mathematics as tools and as a description and study of the regularity of the real world.

Faddeev formulated one of his fundamental methodological and pedagogical principles in the preface to his book Lektsii po algebre (Lectures on algebra):

"I consider that they (that is, abstract concepts) must be introduced to the extent that their introduction succeeds in stimulating in the students the need to generalize or, at least, a realization that it is possible to illustrate sufficiently general concepts by more concrete material". Faddeev maintained this principle in his books for the middle school "Algebra 6-8" and Elementy vysshei matematiki dlya shkol'nikov (Elements of higher mathematics for school-children).

In Faddeev's course, mathematical formalism as an essential effective tool and instrument of pure mathematical research comes after the corresponding concepts and ideas have been interpreted and developed at an interesting intuitive level. And even the second (formal) level of presentation gives the curious and keen pupil examples of sufficiently rigorous mathematical reasoning and proof. Thus, in Faddeev's works, directed at both pupils and teachers, there is to be found a solution to the complicated task of combining a general educational direction with the special training of pupils to continue with the study of mathematics and the application of it at a professional level.

The lines we have listed do not exhaust all aspects of Faddeev's mathematical activity. His numerous and very brilliant papers on the theory of functions, probability theory, and so on, are well known. His teaching over many years at the University, his general and specialized courses of lectures, always excellently presented, his numerous public speeches, are a splendid example to younger mathematicians. His Kurs lektsii po algebre (Course of lectures on algebra) and Sbornik zadach po vysshei algebre (Collection of problems on higher algebra) written on this basis are rightly considered the best books of their kind.

Dmitrii Konstantinovich Faddeev is a man of the broadest culture and intelligence. He has a wide knowledge and appreciation of classical music and is an outstanding pianist. Discussion with him of a variety of questions is highly esteemed by all his friends, colleagues, and acquaintances.

His pupils and colleagues wish Dmitrii Konstantinovich good health, new success in his research, and happiness.

A.D. Aleksandrov, M.I. Bashmakov. Z.I. Borevich,
V.N. Kublanovskaya, M.S. Nikulin, A.I. Skopin, A.V. Yakovlev

### List of D.K. Faddeev's publications

(The beginning of this list was published in Uspekhi Mat. Nauk 13:1 (1958), 236-238; 23:3 (1968), 193-195; 34:2 (1979), 226-228. = Russian Math. Surveys 23:3 (1968), 169-175, 34:2 (1979), 253-260.

 Parallel calculations in linear algebra, Kibernetika 1977, no. 6, 28-40 (with V.N. Faddeeva). MR 58 # 24881. = Cybernetics 13 (1977), 822-831.

 An interpretation of the supplementary condition in the theory of embedding of fields with an Abelian kernel, Zap. Nauchn. Sem. LOMI 94 (1979); 116-118. MR 81j;12004. = J. Soviet Math. 19 (1982), 1050-1051.

 Estimation of the domain containing the solution of a system of linear algebraic equations. Zap. Nauch. Sem. LOMI 90 (1979), 227-228. MR 81j: 15004. = J. Soviet Math. 20 (1982), 2068-2069.

 Selected chapters in analysis and higher algebra (a textbook), Leningrad State University, 1981 (with B.Z. Vulikh and A.N. Podkorytov).

 Some properties of a matrix that is the inverse of a Hessenberg matrix, Zap. Nauchn. Sem. LOMI 111 (1981), 177-179. MR82i:15006. = J. Soviet Math. 24 (1984), 118-120.

 Basic concepts of algebra. Part 1, Preprint, LOMI, Leningrad 1981.

 Basic concepts of algebra, Part 2, Preprint, LOMI, Leningrad 1981.

 Parallel calculations in linear algebra. II, Preprint, LOMI, Leningrad 1981 (with V.N. Faddeeva).

 Problems in algebra for the 6th class, Reprint, Sci. Research Inst., Acad. Fed. Sci. of the USSR, Moscow 1981.

 On the question of the solution of systems of linear algebraic equations, Preprint, LOMI, Leningrad 1982.

 Leonid Vital'evich Kantorovich (on his 70th birthday), Uspekhi Mat. Nauk 37:3 (1982), 201-209 (with A.D. Aleksandrov, M.K. Gavurin, S.S. Kutateladze, V.L. Makarovich, G.Sh. Rubinshtein, and S.L. Sobolev). MR 83m:01047. = Russian Math. Surveys 37:3 (1982), 229-238.

 Parallel calculations in linear algebra. II, Kibernetika 1982, no. 3, 18-31 (with V.N. Faddeeva). MR 84f:65035. = Cybernetics 18 (1982), 288-304.

 Quadratic equations and equations reducing to quadratics. Problems for pupils in classes 6-8, Reprint, Sci. Research Inst., Acad. Fed. Sci. of the USSR, Moscow 1982.

 On equations of higher degrees. Graphs of relations. Problems for pupils in the 7th class. Reprint, Sci. Research Inst., Acad. Ped. Sci. of the USSR, Moscow 1982.

 Algebra 6-8: Introductory topics, Prosveshchenie, Moscow 1983 (Library for Teachers of Mathematics).

 Zenon Ivanovich Borevich (on his 60th birthday), Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1983, no. 3, 114-115 (with S.M. Ermakov, N.N. Polyakov, and A.V. Yakovlev). MR 84i:01085.

 Lektsii po algebre (Lectures on algebra), Nauka, Moscow 1984. MR 86f:00003.

 Galois theory (in the Mathematics Institute of the Academy of Sciences), Trudy Mat. Inst. Steklov. 168 (1984), 46-71. MR 8Sj:12002. = Proc. Steklov. Math. Inst. 1986, no. 3, 47-73.

 Elements of differential calculus. I, Preprint, LOMI, Leningrad 1984 (with M.S. Nikulin and I.F. Sokolovskii).

 Elements of differential calculus. II, Preprint, LOMI, Leningrad 1984 (with M.S. Nikulin and I.F. Sokolovskii).

 On elements of mathematics in secondary schools, Mat. v Shkole, 1985, no. 6, 46-48 (with N.N. Lyashenko, M.S. Nikulin, and I.F. Sokolovskii).

 On elements of mathematics in secondary schools, Preprint, LOMI, Leningrad 1985 (with N.N. Lyashenko, M.S. Nikulin, and I.F. Sokolovskii).

 Trigonometric functions. I, Preprint, LOMI, Leningrad 1985 (with M.S. Nikulin and I.F. Sokolovskii).

 Trigonometrical functions. II, Preprint, LOMI, Leningrad 1985 (with M.S. Nikulin and I.F. Sokolovskii).

 Elements of differential calculus. IL Preprint, LOMI, Leningrad 1985 (with M.S. Nikulin and I.F. Sokolovskii).

 Elements of integral calculus, Preprint, LOMI, Leningrad 1985 (with M.S. Nikulin and I.F. Sokolovskii).

 On a tangent to the graph of a function, Kvant 1986, no. 3, 17-20.

 Universal and methodic principles in the teaching of higher mathematics in secondary schools, in: Methodological problems in mathematical education, 1987, 63-81 (with N.N. Lyashenko, M.S. Nikulin, and I.F. Sokolovskii).

 On modules for quadratic extensions of Dedekind rings, Zap. Nauchn. Sem. LOMI 160 (1987), Anal. Teor. Chisel i Teor. Funkts. 8, 262, 303. MR 89a: 13020.

 On a class of indeterminate equations of the third degree, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1987, no. 3, 60-61. MR 89c: 11043. = Vestnik Leningrad. Univ. Math. 20:3 (1987), 66-67.

 Text-books on algebra and the beginnings of analysis for classes 6-10, Uspekhi Mat. Nauk 42:2 (1987), 260-261 (with M.S. Nikulin and I.F. Sokolovskii).

 Elements of higher mathematics for school-children, Nauka, Moscow 1987 (with M.S. Nikulin and I.F. Sokolovskii).

 Problems in algebra for classes 6-8, Prosveshchenie, Moscow 1988 (with N.N. Lyashenko, M.S Nikulin, and I.F. Sokolovskii). (Library for Teachers of Mathematics.)

 Fundamental principles of differential calculus. I, The linear function, Kvant 1988, no. 3, 46-50 (with M.S. Nikulin and I.F. Sokolovskii).

 Fundamental principles of differential calculus. 11, Properties of the derivative, Kvant 1988, no. 4, 48-55 (with M.S. Nikulin and I.F. Sokolovskii).

Translated by A. Lofthouse