Download Field Axioms plication operations that satisfy the following ten axioms:

Field Axioms
In Mathematics a Field F is any set endowed with addition and multiplication operations that satisfy the following ten axioms:
For Addition:
A1)
a+b
A2)
(a + b) + c
A3) ∃0 ∈ F such that
a+0
A4) ∃b ∈ F such that
a+b
=
=
=
=
b+a
a + (b + c)
a
0 ⇒ b = −a
=
=
=
=
ba
a(bc)
a
1 ⇒ b = a−1 =
For Muliplication:
M1)
M2)
M3) ∃1 ∈ F such that
M4) ∀a =
6 0, ∃b such that
ab
(ab)c
1a
ab
1
a
Distributive Laws:
D1)
D2)
a(b + c) = ab + ac
(a + b)c = ac + bc
The Real numbers IR are a field. So is IR2 if multiplication is defined the
right way. The Complex plane (number system) is IR2 with this definition
of multiplication.
Notes on notation:
∃ mean ”there exists”
∀ mean ”for all”
3 mean ”such that”