Ancient Babylonians worked with algebraic ideas on tablets. They didn't write their expressions as we would today with the same symbols or signs but the ideas were certainly algebra nonetheless.
One of the first people to introduce symbols and abbreviations when expressing algebraic problems. Wrote "Arithmetica" which was a collection of algebraic problems that was the basis for mathematicians down the road.
Arybhata finds integer solutions to linear equations
Arybhata was able to obtain whole number solutions to linear equations with a method that is fairly similar to the one we still use today.
Brahmagupta and linear equations
Brahmagupta developed a method for solving linear and quadratic equations. He noticed that a quadratic equation yields two solutions.
Given the name "The Father of Algebra." Wrote the book "al-jabr w'al muqābala" which is where the word algebra comes from. Wrote about algebra using words and not numbers because he wanted to focus on a more general way to solve problems, and thus created the abstract mathematical language we use. He also developed a formula for solving quadratic equations. He is arguably the most important mathematicians.
1400 - 1700
Expressions were originally written out in words instead of with numbers and symbols. When symbols first came about, they were used in a shorthand way and there was very little consistency across the board. Different countries and branches of math had their own way of expressing things. From the 1470s in Germany to the introduction of Parentheses indicating groups in the early 1700s, the way people wrote expressions did a complete 180.
Evolution of Symbols
1490 - 1600
Before this time, words were used to express things in algebra, such as the use of "shai" to mean unknown quantity. Many early symbols were just shorthand versions of the words they represented, like p for plus. Late in the 15th century we see these shorthand expressions appear in the work of famous mathematicians. By the early 16th century we see + and - and symbols for square roots and more complex operations come about. The process of how we got to the symbols we use today was a messy one because it was hard to ensure uniformity.
The equal sign "=" was proposed
Robert Recorde proposed the symbol = to be used to mean equal. This was used in England but took a while to catch on elsewhere.
Viete and the use of letters
1560 - 1600
Viete was focused on solving algebraic equations and wanted a way to clarify and generalize them. He decided to use letters to represent constants and unknown numbers.
An equation of degree n will have n roots
Both Descartes and Albert Gerard figured out that if an equation has a degree of n, it will have n roots if we are allowing real, false, and imaginary roots. This helped tidy up the theory of equations.
Descartes and "La Géométrie"
Introduced what we regard as standard algebraic notation today. AKA the use of a,b,c for known values and x,y,z for unknown values. Also explained how any function can be represented on a plane by plotting coordinates.
Fundamental Theorem of Algebra proved
Carl Gauss proves the Fundamental Theorem of Algebra. The theorem tells us that a polynomial of degree n will have exactly n roots. So there are n values for x that make the polynomial equal to zero.
Galois proved there is no general method for solving a polynomial equation with a degree higher than 4
Dedekind made leaps and bounds
1860 - 1900
Dedekind provided a definition for the real numbers. He built on that idea and was able to explain properties and relationships between sets and their similarity.
Discovered a theorem, now known as the Cartwright Theorem.
Jesse Ernest Wilkins Jr
Worked on the Manhattan Project and used his knowledge of mathematics to construct and solve equations that helped carry out the mission.
Majoriee Lee Browne
Studied and worked in linear and matrix algebra and produced many helpful ideas.