field automorphism

Definition 1: Bijective

A bijective map is a map f from set X to a set Y with the property that for every y in Y, there is exactly one x in X such that f(x) = y.

Definition 2: Homomorphism

A homomorphism is a map from one algebraic structure to another of the same type that preserves certain properties.

Definition 3: Isomorphism

An isomorphism is a bijective map f such that both f and its inverse f^-1 are homomorphisms.

Definition 4: Automorphism

An automorphism is an isomorphism from a mathematical object to itself.

Definition 5: Ring Homomorphism

A ring homomorphism is a mapping between two rings which preserves the operations of addition and multiplication.

If R,S are rings and f: is the mapping R → S, the the following properties hold:

(1) f(a+b) = f(a) + f(b) for all a,b ∈ R

(2) f(ab) = f(a)f(b) for all a,b ∈ R

(3) f(1) = 1

Definition 6: Field Automorphism

A field automorphism is a bijective ring homomorphism from a field to itself.

References

• "Automorphism", Wikipedia.org
  
• "Bijection", Wikipedia.org
• "Homomorphism", Wikipedia.org
  
• "Isomorphism", Wikipedia.org
• "Ring Homomorphism", Wikipedia.org