Wiki finite field：高级算法 (Fall 2022)Finite Field Basics

Let [math]\displaystyle{ S }[/math] be a set, closed under binary operations [math]\displaystyle{ + }[/math] (addition) and [math]\displaystyle{ \cdot }[/math] (multiplication). It gives us the following algebraic structures if the corresponding set of axioms are satisfied.

Structures Axioms Operations field commutativering ring abeliangroup group monoid semigroup 1. Addition is associative: [math]\displaystyle{ \forall x,y,z\in S, (x+y)+z= x+(y+z). }[/math] [math]\displaystyle{ + }[/math] 2. Existence of additive identity 0: [math]\displaystyle{ \forall x\in S, x+0= 0+x=x. }[/math] 3. Everyone has an additive inverse: [math]\displaystyle{ \forall x\in S, \exists -x\in S, \text{ s.t. } x+(-x)= (-x)+x=0. }[/math] 4. Addition is commutative: [math]\displaystyle{ \forall x,y\in S, x+y= y+x. }[/math] 5. Multiplication distributes over addition: [math]\displaystyle{ \forall x,y,z\in S, x\cdot(y+z)= x\cdot y+x\cdot z }[/math] and [math]\displaystyle{ (y+z)\cdot x= ...