Matrix constructions of finite fields

There are finite fields other than those fields , prime, already introduced. This section will introduce some explicit examples. Later in this chapter, we will see how they are constructed more generally.

Example 2.1.19   In this example, we construct a field having elements. Let

where each matrix entry is considered as an element of , then is a field under the operations of matrix addition and multiplication. It has characteristic .

Example 2.1.20   If we let

where each matrix entry is considered as an element of , then is a field. It has characteristic .

Example 2.1.21   Let

This is a field with elements. It has characteristic .

We shall later explain (see §2.7) how these come about in general.

Exercise 2.1.22   Verify the following factorizations in .

(a) ,

(b) ,

(c) .

Are there integers , not both equal to zero, such that ?

Exercise 2.1.23   Show that the field has characteristic . Here is a prime.

Exercise 2.1.24   Write . If , show .

Exercise 2.1.25 (a)   Find all prime elements of with .

(b) Plot these primes as the points on the -plane.

Exercise 2.1.26   Write the addition and multiplication tables for the set in Example 2.1.19. Using these tables, check that is a field of characteristic .

Exercise 2.1.27   Check that is a field.

Exercise 2.1.28   (Assumes linear algebra 2.4) If is, as an -vector space, finite dimensional with dimension then we say that the degree of over is .

Check that is degree over .

Exercise 2.1.29   Suppose , where . Show .

Exercise 2.1.30   Let be the field constructed in Example 2.1.16, with addition and multiplication mod . Compute

(a) (in terms of the basis ),

(b) (in terms of the basis ),

(c) (in terms of the basis ).

Exercise 2.1.31   Let be the field constructed in Example 2.1.16, with addition and multiplication mod . Find the multiplication table for the group and the addition table for the group . Check that is a field.

Exercise 2.1.32   Check that the set in Example 2.1.20 is a field of characteristic .

Exercise 2.1.33   Consider the field in Example 2.1.21. Find a matrix such that

is the set of all elements in except for the 0 matrix.

Exercise 2.1.34   Check that the set in Example 2.1.21 is a field of characteristic .

Exercise 2.1.35   Consider the field in Example 2.1.21. Let . Show that

is the set of all elements in except for the 0 matrix.

Exercise 2.1.36   Try to construct a field of characteristic having elements, by modifying Example 2.1.21.

Exercise 2.1.37   Complete the the addition and multiplication tables for :

 0 0

 0 0
Check that is a field (check all the axioms).

Exercise 2.1.38   Verify the following in .

(a) ,

(b) ,

(c) .

Exercise 2.1.39   Write in the form , for some .

Exercise 2.1.40 (a)   Write down the elements in

closest to 0 . Plot them on the real number line.

(b) Write down the elements in

closest to 0 . Plot them on the real number line. Can you find an such that ?

Exercise 2.1.41   Write . If , show .

Exercise 2.1.42 (a)   Find all prime elements of with .

(b) Plot these primes as the points on the -plane.

Exercise 2.1.43   Let . (a) Write the addition table of .

(b) Write the multiplication table of .

Using these tables, show that

(c) is a field,

(d) has characteristic .

Exercise 2.1.44   Let . (a) Write the addition table of .

(b) Write the multiplication table of .

Using these tables, show that

(c) is a field,

(d) has characteristic .

Exercise 2.1.45   Show that is a field, if is a prime. (Hint: Use Proposition 1.9.2.)

Exercise 2.1.46   Show that is not a field, if is not a prime. (Hint: Use Proposition 1.9.1.)

The following exercise refers to Exercise 2.1.28.

Exercise 2.1.47
1. Show that is an extension field of . Is a finite dimensional vector space over ? (In other words, is the degree finite?) If so, find its degree.
2. Show that is an extension field of . Is the degree of finite (i.e., is a finite dimensional vector space over )? If so, find its degree.

Exercise 2.1.48   Let be a finite non-empty set. Let denote the set of all subsets of . Let set-theoretic intersection denote multiplication'' and let set-theoretic union denote addition''. Show that is a ring with these operations.

Exercise 2.1.49 (a)   Show that , , for any belonging to a field .

(b) Show that the cancellation law'' holds: if and then .

david joyner 2008-04-20