Fields in action
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This fundamental algebraic structure is "modeled" on the properties of the real numbers: it is a set containing at least two elements, equipped with two binary operations, traditionally denoted by + (addition) and × (multiplication), that allow us to form additive and multiplicative inverses. This lets us define subtraction and division (by any nonzero element), subject to the usual algebraic rules, notably the commutativity of the two binary operations and the distributivity of × over +.
Thus, the set of natural numbers is not a field, because subtraction cannot be defined within it (for example, 0 − 1 does not belong to ?). Nor is the set of integers a field, because division cannot be defined within it (indeed, 1 / 2 does not belong to ?). But the set ? of rational numbers, the set of real numbers and also ?, the set of complex numbers, are fields. And there are many others…
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The vector space of polynomials
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