Linear Algebra, 2020a

Topic
Fields
Computations involving only integers or only rational numbers are much easier
than those with real numbers. Could other algebraic structures, such as the
integers or the rationals, work in the place of R in the definition of a vector
space?
If we take “work” to mean that the results of this chapter remain true then
there is a natural list of conditions that a structure (that is, number system)
must have in order to work in the place of R. Afield is a set F with operations
‘+’ and ‘·’ such that
(1) for any a, b ∈ F the result of a + b is in F, and a + b = b + a, and if c ∈ F
then a +(b + c) =(a + b)+c
(2) for any a, b ∈ F the result of a · b is in F, and a · b = b · a, and if c ∈ F then
a · (b · c) =(a · b) · c
(3) if a, b, c ∈ F then a · (b + c) =a · b + a · c
(4) there is an element 0 ∈ F such that if a ∈ F then a + 0 = a, and for each
a ∈ F there is an element −a ∈ F such that (−a)+a = 0
(5) there is an element 1 ∈ F such that if a ∈ F then a · 1 = a, and for each
element a ≠ 0 of F there is an element a −1 ∈ F such that a −1 · a = 1.
For example, the algebraic structure consisting of the set of real numbers
along with its usual addition and multiplication operation is a field. Another
field is the set of rational numbers with its usual addition and multiplication
operations. An example of an algebraic structure that is not a field is the integers,
because it fails the final condition.
Some examples are more surprising. The set B = {0, 1} under these operations:
+ 0 1
0 0 1
1 1 0
· 0 1
0 0 0
1 0 1
is a field; see Exercise 4.