[Math Lair] Fields

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The four basic arithmetical operations are addition, subtraction, multiplication, and division. These operations are sufficient to perform most of our everyday calculations. One might wonder what kind of number system is sufficient to allow us to carry out such arithmetical operations. Note that many systems are not sufficient. For example, the set of the natural numbers supports addition and multiplication fully, but when you subtract or divide a natural number from or by another, the result is often not a natural number (for example, 5/2 = 2.5 is not a natural number, nor is 3 − 5 = −2).

We can express this "deficiency" in the natural numbers by using the concepts of additive and multiplicative inverses. The additive inverse of a number is the number that, when added to the number, produces zero. The multiplicative inverse of a number is the number that, when multiplied by the number, produces one.

A system of numbers wherein every number has an additive inverse and every number except for zero has a multiplicative inverse, and also obeys some basic mathematical laws, is called a field. The rational numbers are an example of a field.

A field is a type of ring; every field is a ring, but not every ring is a field. A field is a commutative ring with unity with at least two elements, where every element except the identity element has an inverse with respect to the second operation (in terms of the rational numbers, the second operation is multiplication, and the inverse of the second operation is division).