Field Structure and Order Structure

Field Structure #

It is an abelian group under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.

Order Structure #

A field S is an ordered field if it satisfies

• Law of trichotomy
• Transitivity
• Compatibility of Order Relation with Addition&Multiplication Composition
  • a->b, a+c->b+c
  • a->b, ac->bc

An order structure can be partially ordered or totally ordered