**A Multicultural History of Algebra**

**V. Frederick Rickey**

To be presented at a festival entitled "Crossing Borders: Globalization in the Arts, Sciences, and Society," SUNY Potsdam, March 31 to April 3, 2004.

Abstract: Mathematics always crosses borders. In fact it seems to recognize no international borders. While some specific bits of mathematics are special to a particular culture, most of mathematics is independent of culture, common to all cultures. To argue this theme we consider the example of the solution of polynomial equations. This topic has a long and interesting history that meanders across many borders. We shall begin in Mesopotamia, move to the Aral Sea, then to Persia, Italy, Germany, Norway, France, and finally to the United States. We will mention a host of interesting characters and describe, for a general audience, the mathematics they did.

Thoughts about what to include. This is probably too much.

Mesopotamians could solve linear and quadratic equations.

Map of the region. 10000 archeological sites. Picture of a tablet. Example problem.

Should the Greeks get a nod? Did they really solve equations?

Show a proposition from Euclid and ask them if this is an equation.

Al-Khwarizmi.

Etymology of his name. Map.

Origin of the word Algebra. Picture of barber pole. Joke: WMD (Math Intell Fall 2003, p. 86).

Completing the square. Picture of Latin text.

The six types of quadratic equations. Abstraction makes our life easier.

Umar Khayyami on the cubic. Poem.

Cardano and the cubic. There was a long tradition before this of trying to solve the cubic.

Picture of him. Title page of Ars Magna.

Picture of blocks. How to solve the equation. Poem.

Record and the equals sign.

Title page of Whetstone of Witte. Explain the trilingual pun.

The first equations.

Gauss on the Fundamental Theorem of Algebra.

Abel shows the quintic has no algebraic solution.

Picture of Abel on currency. New stamps. Cathedral School library.

Map showing Finoy, Oslo, Gottingen, Paris. His travels.

Crelle's journal. Pun on the name. Picture at Gottingen.

Goat wallpaper. His tomb.

Galois looks at what equations can be solved.

Picture. Duel.

A page from his letter to Auguste Chevalier.

David Dummit at Vermont who is on the Mathematica poster (explicit roots for solvable quintics). Check MathSciNet for other recent work.

Mathematica solves a cubic. You should recognize the result for what it is.

**References:**

- Isabella Bashmakova and Galina Smirnova,
*The Beginnings & Evolution of Algebra*, MAA, 2000. - J. L. Berggren,
*Episodes in the Mathematics of Medieval Islam*. - V. S. Varadarajan,
*Algebra in Ancient and Modern Times*, American Mathematical Society and Hindustan Book Agency, 1998. Reviewed by Ed Sandifer. - B. L. van der Waerden,
*A History of Algebra*, Springer-Verlag, 1985.