Matrix constructions of finite fields

There are finite fields other than those fields ${\mathbb{F}}_p$ , $p$ 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 $4$ elements. Let

$$F=\{ \left( \begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\rig... ...\left( \begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right) \},$$

where each matrix entry is considered as an element of ${\mathbb{F}}_2$ , then $F$ is a field under the operations of matrix addition and multiplication. It has characteristic $2$ .

Example 2.1.20   If we let

$$\begin{array}{c} F=\{ \left( \begin {array}{ccc} 0&0&0\ \no... ...\ \noalign{\medskip }1&0&1\end {array} \right) \}, \end{array}$$

where each matrix entry is considered as an element of ${\mathbb{F}}_2$ , then $F$ is a field. It has characteristic $2$ .

Example 2.1.21   Let

$$\begin{array}{c} F=\{ \left(\begin {array}{cc} 0&0\ \noalig... ...1\ \noalign{\medskip }4&0\end {array} \right) \}. \end{array}$$

This is a field with $25$ elements. It has characteristic $5$ .

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

Exercise 2.1.22   Verify the following factorizations in ${\mathbb{Z}}[i]$ .

(a) $5=(2+i)(2-i)=(1+2i)(1-2i)$ ,

(b) $13=(2+3i)(2-3i)=(3+2i)(3-2i)$ ,

(c) $17=(4+i)(4-i)=(1+4i)(1-4i)$ .

Are there integers $a,b$ , not both equal to zero, such that $7=(a+bi)(a-bi)$ ?

Exercise 2.1.23   Show that the field $\mathbb{F}_p =\mathbb{Z}/p\mathbb{Z}$ has characteristic $p$ . Here $p$ is a prime.

Exercise 2.1.24   Write $N(a+ib)=a^2+b^2$ . If $r,s\in {\mathbb{Z}}[i]$ , show $N(rs)=N(r)N(s)$ .

Exercise 2.1.25 (a)   Find all prime elements $p=a+ib$ of $\mathbb{Z}[i]$ with $a^2+b^2\leq 25$ .

(b) Plot these primes $a+ib$ as the points $(a,b)$ on the $(x,y)$ -plane.

Exercise 2.1.26   Write the addition and multiplication tables for the set $F$ in Example 2.1.19. Using these tables, check that $F$ is a field of characteristic $2$ .

Exercise 2.1.27   Check that $\mathbb{Q}(\sqrt{2})$ is a field.

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

Check that $\mathbb{Q}(\sqrt{2})$ is degree $2$ over $\mathbb{Q}$ .

Exercise 2.1.29   Suppose $a=b\cdot c^2$ , where $b,c\in \mathbb{Z}$ . Show $\mathbb{Q}(\sqrt{a})=\mathbb{Q}(\sqrt{b})$ .

Exercise 2.1.30   Let $F$ be the field constructed in Example 2.1.16, with addition and multiplication mod $3$ . Compute

(a) $(1+\sqrt{2})^{-1}$ (in terms of the basis $e_1,e_2$ ),

(b) $(1-3\sqrt{2})/(1+\sqrt{2})$ (in terms of the basis $e_1,e_2$ ),

(c) $(1-3\sqrt{2})^{-1}\cdot (1+\sqrt{2})$ (in terms of the basis $e_1,e_2$ ).

Exercise 2.1.31   Let $F$ be the field constructed in Example 2.1.16, with addition and multiplication mod $3$ . Find the multiplication table for the group $(F^\times, \cdot)$ and the addition table for the group $(F,+)$ . Check that $F$ is a field.

Exercise 2.1.32   Check that the set $F$ in Example 2.1.20 is a field of characteristic $2$ .

Exercise 2.1.33   Consider the field $F$ in Example 2.1.21. Find a matrix $A\in F$ such that

$$\{A^i \vert 0\leq i \leq 5\}$$

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

Exercise 2.1.34   Check that the set $F$ in Example 2.1.21 is a field of characteristic $5$ .

Exercise 2.1.35   Consider the field $F$ in Example 2.1.21. Let $A=\left( \begin {array}{cc} 0&1\ \noalign{\medskip }4&1\end {array} \right)$ . Show that

$$\{A^i \vert 0\leq i \leq 23\}$$

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

Exercise 2.1.36   Try to construct a field of characteristic $3$ having $9$ elements, by modifying Example 2.1.21.

Exercise 2.1.37   Complete the the addition and multiplication tables for $\mathbb{F}_2=\{0,1\}$ :

$+$	0	$1$
0
$1$

$\cdot$	0	$1$
0
$1$

Check that $\mathbb{F}_2$ is a field (check all the axioms).

Exercise 2.1.38   Verify the following in ${\mathbb{Z}}[\sqrt{2}]$ .

(a) $2=(2-\sqrt{2})(2+\sqrt{2})=\sqrt{2}\sqrt{2}$ ,

(b) $-1=(1-\sqrt{2})(1+\sqrt{2})$ ,

(c) $(1+\sqrt{2})^{-1}=-1+\sqrt{2}$ .

Exercise 2.1.39   Write $(1+\sqrt{2})^{-5}$ in the form $m+\sqrt{2}n$ , for some $m,n\in \mathbb{Z}$ .

Exercise 2.1.40 (a)   Write down the $10$ elements in

$$\{m+n\sqrt{2} \vert m,n\in\mathbb{Z}\}$$

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

(b) Write down the $10$ elements in

$$I=\{2m+n\sqrt{2} \vert m,n\in\mathbb{Z}\}$$

closest to 0 . Plot them on the real number line. Can you find an $r\in \mathbb{Z}[\sqrt{2}]$ such that $I=r\mathbb{Z}[\sqrt{2}]$ ?

Exercise 2.1.41   Write $N(a+\sqrt{2}b)=a^2-2b^2$ . If $r,s\in {\mathbb{Z}}[\sqrt{2}]$ , show $N(rs)=N(r)N(s)$ .

Exercise 2.1.42 (a)   Find all prime elements $p=a+\sqrt{2}b$ of $\mathbb{Z}[\sqrt{2}]$ with $a^2+b^2\leq 25$ .

(b) Plot these primes $a+\sqrt{2}b$ as the points $(a,b)$ on the $(x,y)$ -plane.

Exercise 2.1.43   Let $F=\mathbb{Z}/3\mathbb{Z}$ . (a) Write the addition table of $F$ .

(b) Write the multiplication table of $F^\times = F-\{0\}$ .

Using these tables, show that

(c) $F$ is a field,

(d) $F$ has characteristic $3$ .

Exercise 2.1.44   Let $F=\mathbb{Z}/5\mathbb{Z}$ . (a) Write the addition table of $F$ .

(b) Write the multiplication table of $F^\times = F-\{0\}$ .

Using these tables, show that

(c) $F$ is a field,

(d) $F$ has characteristic $5$ .

Exercise 2.1.45   Show that $\mathbb{F}_p$ is a field, if $p$ is a prime. (Hint: Use Proposition 1.9.2.)

Exercise 2.1.46   Show that $\mathbb{Z}/n\mathbb{Z}$ is not a field, if $n$ 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 $\mathbb{C}$ is an extension field of $\mathbb{R}$ . Is $\mathbb{C}$ a finite dimensional vector space over $\mathbb{R}$ ? (In other words, is the degree finite?) If so, find its degree.
2. Show that $\mathbb{R}$ is an extension field of $\mathbb{Q}$ . Is the degree of $\mathbb{R}/\mathbb{Q}$ finite (i.e., is $\mathbb{R}$ a finite dimensional vector space over $\mathbb{Q}$ )? If so, find its degree.

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

Exercise 2.1.49 (a)   Show that $a+(-a)=0$ , $a \cdot 0=0$ , for any $a$ belonging to a field $F$ .

(b) Show that the ``cancellation law'' holds: if $ac=bc$ and $c\not= 0$ then $a=b$ .