Field Axioms

Field Axioms of R

with two binary operations and satisfying the following properties

1. Closed under addition:
2. Commutative:
3. Associative:
4. Additive identity:
5. Additive inverse:
6. Closed under multiplication:
7. Commutative:
8. Associative:
9. Multiplicative identity:
10. Multiplicative inverse:
11. Multiplication is distributive over addition:

Required proofs

The below mentioned propositions can and should be proven using the above-mentioned axioms. .

• Hint: Start with
• Additive identity () is unique
• Multiplicative identity () is unique
• Additive inverse () is unique for a given
• Multiplicative inverse () is unique for a given

Field

Any set satisfying the above axioms with two binary operations (commonly and ) is called a field. Written as:

Field or Not?

	Is field?	Reason (if not)
	True
	False	Axiom 11 is invalid
	False	Multiplicative inverse doesn’t exist
	True
	False
Boolean algebra	False	Additive inverse doesn’t exist
	True
	True
	False	Multiplicative inverse doesn’t exist