Field (mathematics)
In mathematics a field is a certain kind of algebraic structure. In a field you can add ([math]\displaystyle{ x+y }[/math]), subtract ([math]\displaystyle{ x-y }[/math]), multiply ([math]\displaystyle{ x\cdot y }[/math]) and divide ([math]\displaystyle{ x/y }[/math]) two numbers (with division only allowed if [math]\displaystyle{ y }[/math] is not equal to zero). A field is a special ring, in which you can divide.
Rules
As the field is always a ring, it consists of a set (represented here with the letter R) with two operations: addition (+) and multiplication (•). The properties of a ring are:
- Closure: If an operation is used on any elements in the ring, the element that is formed will also be part of the ring.
- For all a, b in R, the result of the operation a + b is also in R.
- For all a, b in R, the result of the operation a • b is also in R.
- Additive Identity element: One element of the ring is special. It is called the additive identity element (or zero). If addition is used with the identity element and another element, the other element will not change.
- There exists an element 0 in R, such that for all elements a in R, the equation 0 + a = a + 0 = a holds.
- Associativity of Addition: When addition is done many times, it does not matter how it is grouped, the result will be the same.
- For all a, b and c in R, the equation (a + b) + c = a + (b + c) holds.
- Commutativity of addition: When addition is done, it does not matter which element is on the right and left, the result will be the same.
- For all a and b in R, the equation a + b = b + a holds.
- Additive Inverse element: Every element in the ring has another element in the ring when addition is performed between them, the result is the additive identity element. This is known as its additive inverse.
- For each a in R, there exists an element -a in R such that a + (-a) = 0, where 0 is the additive identity element.
- Associativity of multiplication: When multiplication is done many times, it does not matter how it is grouped, the result will be the same.
- For all a, b and c in R, the equation (a • b) • c = a • (b • c) holds.
- Distribution: Addition and multiplying mix together:
- For all a, b and c in R, the equation (a + b) • c = (a • c) + (b • c) holds.
- For all a, b and c in R, the equation a • (b + c) = (a • b) + (a • c) holds.
A field is a ring with these rules:
- Commutativity of multiplication: When multiplication is done, it does not matter which element is on the right and left, the result will be the same.
- For all a and b in R, the equation a • b = b • a holds.
- Multiplicative Identity element: One element of the ring is special. It is called the multiplicative identity element (or unit). If multiplication is used with the identity element and another element, the other element will not change.
- There exists an element 1 in R, such that for all elements a in R, the equation 1 • a = a • 1 = a holds.
- Multiplicative Inverse element: Every element element in the ring apart from the zero element (the additive identity) has an inverse. Multiplying an element by its inverse will give
- For each a in R, apart from 0, there exists an element a-1 such that a • a-1 = a-1 • a = 1 holds.
- Integral domain: Multiplying by zero is the only multiplication that produces zero.
- For all a and b in R, if a • b = 0 then either a = 0 or b = 0 must be true.
Another way of describing a field is this:
- The set R using the operation + is an abelian (commutative) group with the zero element as its identity element.
- The set R, without the zero element, using the operation • is also an abelian group, with the unit element as its identity element.
- The field also satisfies the distribution rules, listed above.
Examples of fields
Examples for fields are
- the real numbers [math]\displaystyle{ \mathbb{R} }[/math]
- the rational numbers [math]\displaystyle{ \mathbb{Q} }[/math]
- the complex numbers [math]\displaystyle{ \mathbb{C} }[/math]
The integers [math]\displaystyle{ \mathbb{Z} }[/math] are not a field, because you cannot always divide without a remainder.