Theory of equations
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In
Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics, to avoid confusion between old and new meanings of "algebra".
History
Until the end of the 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear polynomial equation in a single
The case of higher degrees remained open until the 19th century, when Paolo Ruffini gave an incomplete proof in 1799 that some fifth degree equations cannot be solved in radicals followed by Niels Henrik Abel's complete proof in 1824 (now known as the Abel–Ruffini theorem). Évariste Galois later introduced a theory (presently called Galois theory) to decide which equations are solvable by radicals.
Further problems
Other classical problems of the theory of equations are the following:
- Linear equations: this problem was solved during antiquity.
- algorithms) to solve these systems remains an active subject of research now called linear algebra.
- Finding the integer solutions of an equation or of a system of equations. These problems are now called Diophantine equations, which are considered a part of number theory (see also integer programming).
- Systems of polynomial equations: Because of their difficulty, these systems, with few exceptions, have been studied only since the second part of the 19th century. They have led to the development of algebraic geometry.
See also
- Root-finding algorithm
- Properties of polynomial roots
- Quintic function
References
Further reading
- Uspensky, James Victor, Theory of Equations (McGraw-Hill), 1963
- Dickson, Leonard E., Elementary Theory of Equations (Internet Archive), originally 1914 [1]
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