Fields
Fields extend the concept of groups to include another operation. Now, mathematicians end up defining fields with the familiar symbols of ⋅ and +, and they even call them multiplication and addition, but hopefully by now, you can see that in abstract algebra, these are just general terms that can mean anything. So, without further ado, let's define a field.
A field is a set (denoted by S) and two operations (+ and ⋅) that we will notate as {S, +, ⋅}, which follows these rules:
- ⟨S, +⟩ is an Abelian group with the identity element e = 0.
- If you exclude the number 0 from the set S to produce a new set S', then ⟨S', ⋅ ⟩ is an Abelian group with the identity element e = 1.
- For the rule of distributivity, let a, b, c ∈ S. Then, a ⋅ (b + c) = a ⋅ b + a ⋅ c.
The set of real numbers ℝ, with the operations of addition and multiplication, is the most obvious...
