A breif history

Algorithms for algebraic calculations have been developed since the Antique times

• The Euclidean Algorithm for computing GCDs (Euclide, around 300 B.C.)
• The method of Erastothene for testing primality of numbers (around 200 B.C.)
• Chinese Remaindering Algorithm (Sun-Tsu, 100 A.C)
• etc.

An intense algorithmic activity for solving equations since the Renaissance

• Descates Rule for isolating real roots of univariate polynomials (Descartes 1596 - 1650)
• The complexity analysis of the Eucliddean Algorithm by Lamé (1845)
• Newton iteration for solving equations (1669)
• Galois Theory for deciding when univariate equations can be solved by radicals [Esc01]
• First algorithm for factoring univariate polynomials by Kronecker [Kro97b,Kro97a]
• The method of Gauss for multiplying complex numbers, see also Karatsuba's trick [KO63].
• etc.

The discovery of fast algorithms since the introduction of computers

• Karatsuba's trick for univariate polynomial multiplication [KO63],
• FFT-based univariate polynomial multiplication (Cooley and Tukey, 1965),
• First algorithm and its implementation of polynomial system solver [Buc65]
• Strassen's method for matrix multiplication [Str69]
• First efficient algorithm for factoring polynomials [Ber67,Zas69,LHWLL82]
• Fast algorithm for division (Cook, 1966) (Sieveking, 1972)
• Fast Chinese Remaindering Algorithm (Borodin & Moenck)
• Fast algorithms for GCD computations (Knuth, 1970), (Schönage, 1971) and others