The author is the most outstanding personality in the brilliant school of mathematicians from the USSR in algebraic number theory, algebraic geometry and group theory. True to the spirit of 20th century mathematics, he always has been on the alert for connections between these three fields of research, and has been able to make a remarkable use of them.
This volume contains about half of his 92 published papers, all of them in English, those originally in Russian having been translated; this provides easy access for papers some of which are in not easily accessible journals.
The author’s early work was concerned with a problem still unsolved in general, the description of normal extensions of a number field or a local field, for non-abelian Galois groups. He came to the attention of the mathematical world by solving in 1946 the following particular case: The fundamental field \(k\) is an extension of the \(p\)-adic field \(\mathbb Q_p\) of degree \(n_0\), \(k\) does not contain the \(p\)-th roots of unity, and one looks at the Galois extensions of k whose degree over \(k\) is a power of \(p\). He showed that these extensions are in one to one correspondence with the normal subgroups \(N\) of the free group \(S\) with \(n_0+1\) generators, such that \(S/N\) is a \(p\)-group.
The author published many more papers on extensions of number fields satisfying various conditions. In 1954, he showed that there are infinitely many extensions whose Galois group is a given solvable group. Then, in 1964, he turned to the consideration of finite or infinite extensions, whose Galois group is a pro-\(\ell\)-group for a given prime \(\ell\), and which are ramified only at the primes of a given finite set \(S\). If \(G_S\) is the Galois group of the maximal extension having these properties, he focused the study of the group on the computation of the minimal number \(d(G_S)\) of generators of \(G_S\) and of the minimal number \(r(G_S)\) of relations between these generators. In particular, \(G_S\) is infinite if there exists a sequence of finite \(\ell\)-groups \(G_i\) for which \(d(G_i)\) and \(r(G_i)-d(G_i)\) tend to infinity. Then, in a joint paper with E. S. Golod, using homological algebra, they were able to prove that \(r(G)\geq (d(G)-1)^2/4\) for all finite \(\ell\)-groups; for \(S=\emptyset\) this enabled them to give examples of number fields which have infinite unramified extensions, thus solving in particular the famous “class field tower” problem, which had been open since Hilbert.
After 1960, much of the author’s research was devoted to algebraic geometry, in which he was one of the first to use the theory of schemes. He published a well-known textbook on the subject, and, with several co-authors, another more specialized book on algebraic surfaces; two chapters of that last book are reproduced in this volume. He has been particularly interested in algebraic varieties over finite fields or algebraic number fields. One of his conjectures on good reduction of varieties defined over number fields was shown by A. Parshin in 1970 to imply the Mordell conjecture, and the latter was finally proved by G. Faltings in 1983 as a consequence of a generalization of Shafarevich’s conjecture.
In collaboration with A. Kostrikin, the author has discovered an unsuspected connection between Élie Cartan’s results of simple “infinite Lie pseudo-groups” and simple Lie algebras over algebraically closed fields of characteristic \(p>0\). The modern interpretation of Cartan’s pseudogroups consists in Lie subalgebras of the Lie algebra of derivations of a ring \(k[[x_1,\dots,x_n]]\) of formal power series over a field \(k\); the usual theory assumes that k has characteristic 0, but the definitions are valid for a field \(k\) of characteristic \(p>0\). Kostrikin and the author observed that for such fields the Lie algebras corresponding to the simple pseudogroups are not simple anymore, but have quotients by naturally defined ideals, which turn out to be simple Lie \(p\)-algebras of finite dimension over \(k\). It was later shown that these Lie algebras (which had been earlier defined by other methods) are for \(p>5\) the only simple Lie \(p\)-algebras different from those defined by Chevalley and corresponding to the classical simple Lie algebras.