math111_logo Appendix: Field

1. Number system and field

In developing the theory of linear algebra, we almost always use real numbers R as the scalar (coefficients, coordinates, entries, etc). In fact, we may also use other number systems such as

rational numbers Q: numbers of the form m/n, where m, n are integers;

complex numbers C: numbers of the form a + ib, where a, b are real numbers and i = √-1.

Two things need to be explained in order to justify the use of number systems other than the real numbers in linear algebra.

For the first question, a general philosophical answer is that linear algebra is the language used to model linear structures. The linear structures not only appear over real number, but also appear over other number systems. We would like to emphasis that the complex linear structures are especially important. One key advantage of C over R is that C is algebraically closed, meaning that any polynomial has complex roots. As a result, complex numbers (but not real numbers) provide solutions to many problems. For example, it is necessary to include complex numbers in order to gain deeper understanding of eigenvalue and eigenvectors.

For the second question, we may revisit all the theories we have developed and find out that the four usual operations of numbers (summation, substraction, multiplicaiton, division) are all the ones we needed. Moreover, we also used the usual properties of these operations, such as a + b = b + a, a(b + c) = ab + ac. This leads to the following generalization of real numbers.

A field is a set F, with two operations

addition
a + b: F×FF
multiplication
ab: F×FF

satisfying the following properties

commutative
a + b = b + a, ab = ba
associative
(a + b) + c = a + (b + c), (ab)c = a(bc)
distributive
a(b + c) = ab + ac
zero
There is a special element 0 ∈ F such that 0 + a = a
nagative
For any aF there is an element -a such that a + (-a) = 0
unit
There is a special element 1∈ F such that 1a = a
inverse
For any aF - {0} there is an element a-1 such that aa-1 = 1

The substraction is then a - b = a + (-b), and the division is a/b = ab-1. Note the similarity to the definition of vector spaces.

Example With the usual addition and multiplicaiton, the real numbers form a field R. The rational numbers also form a field Q. The key reason here is that if a and b are rational, then a + b and ab are also rational. This makes Q a subfield of R (compare with the definition of subspace and this result).

The integers Z is not a field because the inverse does not always exist. The positive real numbers R+ is also not a field because zero and negative do not exist.

Example A complex numbers is a number of the form z = a + ib, where i = √-1 is defined by the property

i2 = -1.

The addition and multiplication are defined by the usual method:

(a + ib) + (c + id) = (a + c) + i(b + d),
(a + ib)(c + id) = ac + iad + ibc + i2bd = (ac - bd) + i(ad + bc).

With these operations, we may verify that the collection C of all complex numbers is a field. The inverse is, in particular, given by

(a + ib)-1 = a - ib .
a2 + b2

Example Polynomials do not form a field because the inverse of a polynomial is generally not a polynomial. To get a field out of the polynomials, therefore, we introduce rational functions, which are the quotient of two polynomials. The following are some examples of rational functions

1 , 1 + t2 ,

2 - t + t2

, 1 + t2 , 3 - 5t + 4t4 .
1 + t 1 - t2    1 - t2    1 + t - t2 - t3 1 + 2t + t2 + t3

The addition and multiplication can be done in the usual way:

1 + 1 + t2 =

2 - t + t2

, 1 × 1 + t2 =

2 - t + t2

.
1 + t 1 - t2    1 - t2    1 + t 1 - t2 1 + t - t2 - t3

With the two operations, we get the field R(t) of (real) rational functions.

In general, for any field F, we have the field F(t) of rational functions over F.

Example Let F = {0, 1, 2, 3, 4}. For any integer n, define "n modulo 5" to mean dividing the number by 5 and keeping the remainder only. For example,

13 = 2×5 + 3 → 13 modulo 5 = 2,
-8 = (-2)×5 + 2 → -8 modulo 5 = 2.

Then we define the addition and multiplication in F to be the usual one modulo 5. For example

3 + 4 = 7 modulo 5 = 2, 1 + 3 = 4 modulo 5 = 4;
2×2 = 4 modulo 5 = 4, 3×4 = 12 modulo 5 = 2.

With these operations, F form a field, called finite field of order 5.

The only nontrivial properties to check is the existence of inverse, which is done below,

1×1 = 1 modulo 5 = 1 → 1-1 = 1,
2×3 = 6 modulo 5 = 1 → 2-1 = 3,
3×2 = 6 modulo 5 = 1 → 3-1 = 2,
4×4 = 16 modulo 5 = 1 → 4-1 = 4.

Although the formulae above are somewhat counter-intuitive, we obtain them, however, by strictly following the rules.

The key reason behind the existence of the inverse is the fact that 5 is a prime number. In general, for any prime number p, we may define the finite field of order p in similar way.


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