Modern Algebra
Basic algebraic structures Fields In itself a set is not very useful, being little more than a well-defined collection of mathematical objects. However, when a set has one or more operations (such as addition and multiplication) defined for its elements, it becomes very useful. If the operations satisfy familiar arithmetic rules (such as associativity, commutativity, and distributivity) the set will have a particularly “rich” algebraic structure. Sets with the richest algebraic structure are known as fields. Familiar examples of fields are the rational numbers (fractions a/b where a and b are positive or negative whole numbers), the real numbers (rational and irrational numbers), and the complex numbers (numbers of the form a + bi where a and b are real numbers and i2 = −1). Each of these is important enough to warrant its own special symbol: ℚ for the rationals, ℝ for the reals, and ℂ for the complex numbers. The term field in its algebraic sense is quite different from its use i...