“The First Field”

We all know one field that contains 0,1,2,..., but, logically, there is an earlier field that is defined as follows. We first fill in the addition table, subject to the condition that before we fill in the entry for a+b, we must have already filled in all entries a´+b and a+b´ with a´We then tackle the multiplication table of a field with the given addition. Again, the entries are to be the least possible one´s subject to this requirement; this construction produces a very strange field in which 8 is a fifth root of unity. Amazingly, this field actually has practical applications.

John H. Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life.

Conway is currently Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics at Princeton University. He studied at Cambridge, where he started research under Harold Davenport. He received the Berwick Prize (1971), was elected a Fellow of the Royal Society (1981), was the first recipient of the Pólya Prize (LMS) (1987), won the Nemmers Prize in Mathematics (1998), and received the Leroy P. Steele Prize for Mathematical Exposition (2000) of the American Mathematical Society.