Ernst Steinitz’s studies of the algebraic theory of mathematics provided the basic solution methods for polynomial roots, initiating the methodology and domain of abstract algebra.
Before 1900, algebra and most other mathematical disciplines focused almost exclusively on solving specific algebraic equations, employing only real, and less frequently complex, numbers in theoretical as well as practical endeavors. One result of the several movements contributing to the so-called abstract turn in twentieth century algebra was much-increased technical economy through introduction of symbolic operations, which was accompanied by a notable increase in generality and scope.
Although the axiomatic foundationalism of David Hilbert
Kronecker had unique convictions about how questions on the foundations of mathematics should be treated in practice. In contrast to Richard Dedekind,
The question of finding algebraic roots in general had been of fundamental import since the prior work of Galois, Niels Henrik Abel,
The novelty of the field approach is seen from the introduction to Weber’s contemporaneous paper “Die allgemeinen Grundlagen der Galois’chen Gleichungstheorie” (the general foundations of Galois theory). Weber first proved an important theorem stated by Kronecker, which relates the field of rational numbers to so-called cyclotomic, or Abelian, groups, a subsequently important area of the development of field theory. Weber also established the notion of a “form field,” being the field of all rational functions over a given base field F, as well as the crucial notion of the extension of an algebraic field. Although the main part of Weber’s paper interprets the group of an algebraic equation as a group of permutations of the field of its algebraic coefficients, Weber’s exposition is complicated by many elaborate and incomplete definitions, as well as a premature attempt to encompass all of algebra, instead of only polynomials. In his noted 1893 textbook on algebra, Weber calls F(a) an algebraic field when a is the root of an equation with coefficients in F, equivalent to the definition given by Kronecker in terms of the “basis” set for F(a) over a.
A central concern of Weber and other algebraists was that of extending the idea of absolute value, or valuation, beyond its traditional usage. For example, if F is the field of rational numbers, the ordinary absolute value |a| is the valuation. The theory of general algebraic valuations was originated by Kronecker’s student Kurt Hensel when he introduced the concept of p-adic numbers. In his paper “Über eine neue Begründung der algebraischen Zählen” (1899; on a new foundation of the algebraic numbers), Weierstrass’s method of power-series representations for normal algebraic functions led Hensel to seek an analogous concept for the newer theory of algebraic numbers. If p is a fixed rational prime number and a is a rational number not zero, then a can be expressed uniquely in the form a = (r/s) p
n
, where r and s are prime to p. If ϕ (a) = p
-n
, for a ≠ 0, ϕ (a) is a valuation for the field of rational numbers. For every prime number p, there corresponds a number field, which Hensel called the p-adic field, where every p-adic number can be represented by a sequence.
At this time, the American mathematician Joseph Wedderburn was independently considering similar problems. In 1905, he published “A Theorem on Finite Algebra,” which proved effectively that every algebra with finite division is a field and that every field with a finite number of elements is commutative under multiplication, thus further explicating the close interrelations between groups and fields.
Two years after Hensel’s paper appeared, Steinitz published his major report, “Algebraische Theorie der Körper” (1909; theory of algebraic fields), which took the field concepts of Kronecker, Weber, and Hensel much further. Steinitz’s paper explicitly notes that it was principally Hensel’s discovery of p-adic numbers that motivated his research on algebraic fields. In the early twentieth century, Hensel’s p-adic numbers were considered (by the few mathematicians aware of them) to be totally new and atypical mathematical entities, whose place and status with respect to then-existing mathematics was not known. Largely as a response to the desire for a general, axiomatic, and abstract field theory into which p-adic number fields would also fit, Steinitz developed the first steps in laying the foundations for a general theory of algebraic fields.
Steinitz constructed the roots of algebraic equations with coefficients from an arbitrary field, in much the same fashion as the rational numbers are constructable from the integers (aX = b), or the complex numbers from real numbers (x2
= –1). In particular, Steinitz focused on the specific question of the structure of what are called inseparable extension fields, which Weber had proposed but not clarified. Many other innovative but highly technical concepts, such as perfect and imperfect fields, were also given. Perhaps most important, Steinitz’s paper sought to give a constructive definition to all prior definitions of fields, therein including the first systematic study of algebraic fields solely as “models” of field axioms. Steinitz showed that an algebraically closed field can be characterized completely by two invariant quantities: its so-called characteristic number and its transcendence degree. One of the prior field concepts was also clarified.
Although Steinitz announced further investigations—including applications of algebraic field theory to geometry and the theory of functions—they were never published. Nevertheless, the import and implications of Steinitz’s paper were grasped quickly. It was soon realized that generalized algebraic concepts such as ring, group, and field are not merely formally analogous to their better-known specific counterparts in traditional algebra. In particular, it can be shown that many specific problems of multiplication and division involving polynomials can be simplified greatly by what is essentially the polynomial equivalent of the unique factorization theorem of algebra, developed directly from field theory in subsequent studies.
In 1913, the concept of valuation was extended to include the field of complex numbers. An American algebraist, Leonard Dickson (1874-1957), further generalized these results to groups over arbitrary finite fields. Perhaps most notably, the French and German mathematicians Emil Artin
As noted by historians of mathematics, further recognition and adoption of the growing body of work around Steinitz’s original publication continued. Major texts on modern algebra, such as that published by Bartel Leendert van der Waerden in 1932, already contained substantial treatment of Steinitz’s key ideas. As later pointed out by the “structuralist” mathematicians of the French Nicolas Bourbaki group,
The theory of algebraic fields is not only an abstract endeavor but also, since the late 1940’s, has proven its utility in providing practical computational tools for many specific problems in geometry, number theory, the theory of codes, and data encryption and cryptology. In particular, the usefulness of algebraic field theory in the areas of polynomial factorization and combinatorics on digital computers led directly to code-solving hardware and software such as maximal-length shift registers and signature sequences, as well as error-correcting codes. Together with Noether’s theory of rings and ideals, Steinitz’s field theory is at once a major demarcation between traditional and modern theory of algebra and a strong link connecting diverse areas of contemporary pure and applied mathematics.
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Artin, Emil. Algebraic Numbers and Algebraic Functions. 1951. Reprint. Basel, Switzerland: Gordon and Breach Science Publishers, 1994. Discusses Artin’s work on furthering Steinitz’s field theory. -
Budden, F. J. The Fascination of Groups. Cambridge, England: Cambridge University Press, 1972. Represents modern efforts at elementary and intermediate-level treatments. A unique introductory treatment of groups using numerous examples for art, geometry, and music. -
Dickson, Leonard E. Algebras and Their Arithmetics. 1938. Reprint. Mineola, N.Y.: Dover, 1972. Contains some of the first simplified discussions of Steinitz’s work. -
Kline, Morris. Mathematical Thought from Ancient to Modern Times. Vol. 3. 1972. Reprint. New York: Oxford University Press, 1990. Part of a multivolume survey of developments in mathematics since its beginnings. Chapter 49 describes the emergence of abstract algebra, including Steinitz’s work. -
McEliece, Robert. Finite Fields for Computer Scientists and Engineers. Boston: Kluwer Academic, 1987. Advanced text details the practical applications of the algebraic field theory.
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