Fields

Fields are sets that are closed under addition and multiplication.

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 Z_p or F_p 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.

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