Linear Algebra

Topic: Fields 141
Topic: Fields
Linear combinations involving only fractions or only integers are much easier
for computations than combinations involving real numbers, because computing
with irrational numbers is awkward. Could other number systems, like the
rationals or the integers, work in the place of R in the definition of a vector
space?
Yes and no. If we take “work” to mean that the results of this chapter
remain true then an analysis of which properties of the reals we have used in
this chapter gives the following list of conditions an algebraic system needs in
order to “work” in the place of R.
Definition. A field is a set F with two operations ‘+’ and ‘·’ such that
(1) for any a, b ∈F the result of a + b is in F and
• a + b = b + a
• 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
• 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
• 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
• for each non-0 element a ∈F there is an element a −1 ∈F such that
a −1 · a =1.
The number system comsisting of the set of real numbers along with the
usual addition and multiplication operation is a field, naturally. 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 integer number
system—it fails the final condition.
Some examples are surprising. The set {0, 1} under these operations:
isafield(seeExercise4).
+ 0 1
0 0 1
1 1 0
· 0 1
0 0 0
1 0 1