Chapter 6

Fields

There is astonishing imagination, even in the science of mathematics . . . . We repeat, there was more imagination in the head of Archimedes than in that of Homer.

—Voltaire

Human beings have sought solutions to algebraic equations for centuries. This search has inspired some of the most creative (and important) mathematics imaginable. Suppose that a primitive tribe, motivated by the desire to count things and to tell others the results, has developed a facility with the set img of natural numbers to the point where they can add and multiply. Then they can solve certain equations: for example, img has the unique solution img. However, they declare that, despite the efforts of their finest mathematicians, the equation img has no solution. We, of course, know that they have an inad-equate number supply and are not aware of the existence of the negative integers. To put it another way, they have invented a system img of numbers that is adequate for ordinary counting, but they must invent ...

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