1. Theory of Equations Introduction

2. In the 19th century, the theory of equations acquired its status as an independent mathematical discipline. In the process, NIELS HENRIK ABEL (1802–1829) played an important role.

3. His works on the algebraic unsolvability of the general quintic equation and his penetrating studies of the so-called Abelian equations belong to the first results established within this incipient discipline.

4. ABEL’s researches on the theory of equations were rooted in a tradition comprising the works of LEONHARD EULER (1707–1783), ALEXANDRE-TH´E OPHILE VANDERMONDE (1735–1796), and— in particular— JOSEPH LOUIS LAGRANGE (1736–1813).

5. The Italian PAOLO RUFFINI (1765–1822) had in 1799, working within the same tradition as ABEL, as the first mathematician sought to prove the impossibility of solving the general quintic algebraically. RUFFINI published his investigations on numerous occasions, but his presentations were generally critized for lacking clarity and rigour. Not until 1826 — after ABEL had published his proof of this result (Abel 1824; Abel 1826a) — did ABEL mention RUFFINI’s proofs, and I believe that ABEL had independently obtained his result on the unsolvability of the quintic.

6. In LAGRANGE’s comprehensive study of the solution of equations (Lagrange 1770–1771) originated the idea of studying the numbers of formally distinct values which a rational function of multiple quantities could take when these quantities were permuted.

7. The idea was cultivated into an emerging theory of permutations which AUGUSTINLOUIS CAUCHY (1789–1857) in (1815a) provided with its basic notation and terminology. CAUCHY also established the first important theorem within this theory when he proved a generalization of RUFFINI’s result that no function of five quantities could have three or four different values under permutations of these quantities.

8. ABEL combined the results and terminology of CAUCHY’s theory of permutations with his own investigation of algebraic expressions (radicals). ABEL was led to study such expressions in a natural way, as the study of the “extent” of the class of algebraic expressions would impact on the “expressive power” of algebraic solution formulae. Following his minimal definition of algebraic expressions,

9. ABEL classified these newly introduced objects in a way imposing a hierarchic structure in the class of radicals. The classification enabled ABEL to link algebraic expressions—formed from the coefficients — which occur in any supposed solution formula to rational functions of the equation’s roots.

10. By the theory of permutations, which ABEL had taken over from CAUCHY, he reduced such rational functions to only a few standard forms. Considering these forms individually, ABEL demonstrated—by reductio ad absurdum—that no algebraic solution formula for the general quintic could exist.

11. In the first part of the 19th century, the century-long search for algebraic solution formulae was brought to a negative conclusion: no such formula could be found.

12. To many mathematicians of the late 18th century such a conclusion had been counter-intuitive, but owing to the work and utterings of men like EDWARD WARING (17341–1798), LAGRANGE, and CARL FRIEDRICH GAUSS (1777–1855) the situation was different in the 1820s.

13. ABEL’s proof was also met with criticism and scrutiny
ABEL’s proof was also met with criticism and scrutiny. By and large, though, the criticism was confined to local parts of the proof. The global statement—that the general quintic was unsolvable by radicals was soon widely accepted. In his only other publication on the theory of equations, M´emoire sur une classe particuli`ere d’´equations r´esolubles alg´ebriquement (1829a), ABEL took a different approach.

14. The paper was inspired by ABEL’s own research on the division problem for elliptic functions and GAUSS’ Disquisitiones arithmeticae. In it, ABEL demonstrated a positive result that an entire class of equations characterized by relations between their roots were algebraically solvable.

15. For his 1829 approach, ABEL abandoned the permutation theoretic pillar of the unsolvability proof. Instead, he introduced the new concept of irreducibility and with the aid of the Euclidean division algorithm proved a fundamental theorem concerning irreducible equations.

16. For his 1829 approach, ABEL abandoned the permutation theoretic pillar of the unsolvability proof. Instead, he introduced the new concept of irreducibility and with the aid of the Euclidean division algorithm proved a fundamental theorem concerning irreducible equations. The equations which ABEL studied in (1829a) were characterized by having rational relations between their roots.

17. Using the concept of irreducibility, ABEL demonstrated that such irreducible equations of composite degree, m × n, could be reduced to equations of degrees m and n only one of which might not be solvable by radicals. Furthermore, he proved that if all the roots of an equation could be written as iterations of a rational function, the equation would be algebraically solvable.

18. The most celebrated result contained in ABEL’s M´emoire sur une classe particuli`ere was the algebraic solvability of a class of equations later named Abelian equations by KRONECKER. These equations were characterized by the following two properties:

19. (1) all their roots could be expressed rationally in one root, and
(2) these rational expressions were “commuting” in the sense that if i (x) and j (x) were two roots given by rational expressions in the root x, then ij (x) = ji (x) .

20. By reducing the solution of such an equation to the theory he had just developed, ABEL demonstrated that a chain of similar equations of decreasing degrees could be constructed. Thereby, he proved the algebraic solvability of Abelian equations.
In subsequent sections, ABEL wanted to apply this theory to the division problems for circular and elliptic functions. However, only his reworking of GAUSS’ study of cyclotomic equations was published in the paper.

21. Together, the unsolvability proof and the study of Abelian equations can be interpreted as an investigation of the extension of the concept of algebraic solvability.
On one hand, the unsolvability proof provided a negative result which limited this extension by establishing the existence of certain equations in its complement.

22. On the other hand, the Abelian equations fell within the extension of the concept of algebraic solvability and thus ensured a certain power (or volume) of the concept.
In a notebook manuscript—first published 1839 in the first edition of ABEL’s Oeuvres — ABEL pursued his investigations of the extension of the concept of algebraic solvability.

23. In the introduction to the manuscript, he proposed to search for methods of deciding whether or not a given equation was solvable by radicals. The realization of this program would, thus, have amounted to a complete characterization of the concept of algebraic solvability.

24. ABEL’s own approach to this program was based upon his concept of irreducible equations. In the first part of the manuscript — which appears virtually ready for the press— ABEL gave his definition of this concept. Arguing from the definition, he proved some basic and important theorems concerning irreducible equations.

25. In the latter part of the manuscript — which is, however, less lucid and eventually consists of nothing but equations ABEL reduced the study of algebraic expressions satisfying a given equation of degree μ to the study of algebraic expressions which could satisfy an irreducible Abelian equation whose degree divided μ − 1. This final step, the investigation of solution formulae for such irreducible Abelian equations, was never attempted by ABEL.

26. When ABEL’s attempt at a general theory of algebraic solvability was published in 1839, ´E VARISTE GALOIS (1811–1832) had also worked on the subject. Inspired by the same tradition and exemplar problems as ABEL had been, GALOIS put forth a very general theory with the help of which the solvability of any equation could—at least in principle be decided.

27. GALOIS’ writings were inaccessible to the mathematical community until the middle of the 19th century. His presentational style was brief and — at times — obscure and unrigorous. Many mathematicians of the second half of the 19th century — starting with JOSEPH LIOUVILLE (1809–1882) who first published GALOIS’ manuscripts in 1846 — invested large efforts in clarifying, elaborating, and extending GALOIS’ ideas.

28. In the process, the theory of equations finally emerged in its modern form as a fertile subfield of modern algebra. Part of this evolution was concerned with mathematical styles. The highly computation based mathematical style of the 18th century, to which ABEL had also adhered, was superseeded.

29. The old style had been marked by lengthy, rather concrete, and painstaking algebraic manipulations. This was replaced in the 19th century by a more concept based reasoning, early glipses of which can be seen in ABEL’s use of the concepts of algebraic expression and irreducible equation.

30. End of Chapter 1