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Now we can make one step further. If the elements of a ring, excluding the zero, form an abelian group (with the second operation) then it is a field. For example, write the multiplication table of the remainders of division by 5, and you will see that it satisfies all the requirements for a group: (You will probably have noticed that the group does not contain the number five itself since [5] = [0].)

(Why isn't the set of divisors of six - excluding the zero and under multiplication - a group? That's easy enough, since we have excluded the zero we do not have the result of in our set, so it isn't closed.)

Alex Lopez-Ortiz
Mon Feb 23 16:26:48 EST 1998