Fields
Fields are sets that are closed under addition and multiplication.
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Description
A field is a set for which a pair of binary operations are defined: addition and mutliplication.
The field must be closed under both operations; that is, addition is a map as F + F -> F and multiplication is a map as F × F -> F.
The addition operation must also feature these properties:
commutivity: a + b = b + a
associativity: (a + b) + c = a + (b + c)
There exists a value satisfying an identity property: a + 0 = a.
invertibility: a + (-a) = 0
The multiplication operation must also feature these properties:
commutivity: ab = ba
associativity: (ab)c = a(bc)
There exists a value satisfying an identity property: a1 = a.
invertibility: Formally, because zero is excluded, it is said that ∀ a ∈ F \ {0} aa-1 = 1.
Lastly there js a distributivity property: (a + b)c = ac + bc.
Finite Fields
A non-obvious example of a field is the set {0,1} for which addition is defined (a + b) % 2 and for which multiplication is defined (ab) % 2. Because of the modulus, every possible arithmatic operation is:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 0
0 × 0 = 0
0 × 1 = 0
1 × 1 = 1
The invertibility property of multiplication is the trickiest to satisfy, but due to the modulus, the inverse of 1 is in fact 0. (Recall that 0 is not required to have an inverse.)
Consider then the set {0,1,2} where addition is defined (a + b) % 3 and for which multiplication is defined (ab) % 3. It follows that 1 × 1 = 1 and 2 × 2 = 1, satisfying the invertibility property of multiplication again.
A finite field can be defined by the integers with any prime modulus. They are usually notated Zp or Fp where p is the modulus.
Commutative Rings
There are many sets which satisfy some but not all properties of a field. For example, the set of integers does not contain multiplicative inverses. It is instead a commutative ring.