2500 BC Hieroglyphic numerals used in Egypt.

2400 BC Babylonians begin positional algebraic notation.

600 BC Pythagoreans discuss prime numbers.

250 Diophantus writes Arithmetica, using notation from which modern notation evolved, and insists on exact solutions of equations in integers.

830 al-Khowarizmi writes Al-jabr, a textbook giving rules for solving linear and quadratic equations.

1202 Leonardo of Pisa writes Liber abaci on arithmetic and algebraic equations.

1545 Tartaglia solves the cubic, and Cardano publishes the result in his Ars Magna. Imaginary numbers are suggested.

1580 Viète uses vowels to represent unknown quantities, with consonants for constants.

1629 Fermat becomes the founder of the modern theory of numbers.

1636 Fermat and Descartes invent analytic geometry, using algebra in geometry.

1749 Euler formulates the fundamental theorem of algebra.

1771 Lagrange solves the general cubic and quartic by considering permutations of the roots.

1799 Gauss publishes his first proof of the fundamental theorem of algebra.

1801 Gauss publishes his Disquisitiones Arithmeticae.

1813 Ruffini claims that the general quintic cannot be solved by radicals.

1824 Abel proves that the general quintic cannot be solved by radicals.

1829 Galois introduces groups of substitutions.

1831 Galois sends his great memoir to the French Académie, but it is rejected.

1843 Hamilton discovers the quaternions.

1846 Kummer invents his ideal numbers.

1854 Cayley introduces the multiplication table of a group.

1870 Jordan publishes his monumental Traité, which explains Galois theory, develops group theory, and introduces composition series.

1870 Kronecker proves the fundamental theorem of finite abelian groups.

1872 Sylow presents his results on what are now called the Sylow theorems.

1878 Cayley proves that every finite group can be represented as a group of permutations.

1879 Dedekind defines algebraic number fields, studies the factorization of algebraic integers into primes, and introduces the concept of an ideal.

1889 Peano formulates his axioms for the natural numbers.

1889 Hölder completes the proof of the Jordan–Hölder theorem.

1905 Wedderburn proves that finite division rings are commutative.

1908 Wedderburn proves his structure theorem for finite dimensional algebras with no nilpotent ideals.

1921 Noether publishes her influential paper on chain conditions in ring theory.

1927 Artin extends Wedderburn's 1908 paper to rings with the descending chain condition.

1929 Noether establishes the modern approach to the theory of representations of finite groups.