A field is a set 𝐹 with two binary operations on 𝐹 called:

• addition
• multiplication

Both of these operations must satisfy the field axioms (𝑎, 𝑏, 𝑐 are arbitrary elements of the field 𝐹):

correspondence that associates with each ordered pair of elements of 𝐹 a uniquely determined element of 𝐹.

A binary operation on 𝐹 is a mapping 𝐹×𝐹 → 𝐹, that is, a

Required

(𝑎 + 𝑏) ∊ 𝐹

Required

𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐

Required

𝑎 + 0 = 𝑎

Required

𝑎 + (-𝑎) = 0

Required

𝑎 + 𝑏 = 𝑏 + 𝑎

Required

𝑎 · (𝑏 + 𝑐) = (𝑎 · 𝑏) + (𝑎 · 𝑐)

Distributivity of multiplication over addition

Required

(𝑎 · 𝑏) ∊ 𝐹

Required

𝑎 · (𝑏 · 𝑐) = (𝑎 · 𝑏) · 𝑐

Required

𝑎 · 1 = 𝑎

Required ^-1 = 1

𝑎 · 𝑎

Required

𝑎 · 𝑏 = 𝑏 · 𝑎

Unneeded

N/A

Distributivity of addition over multiplication