Exercise 2.3: Reducible and Irreducible Polynomials

Polynomials of degree $m = 2$, $m = 3$, $m = 4$

Important prerequisites for understanding channel coding are knowledge of polynomial properties.

In this exercise we consider polynomials of the form

$$a(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + \hspace{0.1cm}... \hspace{0.1cm} + a_m \cdot x^{m} \hspace{0.05cm},$$

where for the coefficients $a_i ∈ {\rm GF}(2) = \{0, \, 1\}$ holds $(0 ≤ i ≤ m)$

and the highest coefficient is always assumed to $a_m = 1$.

One refers to $m$ as the "degree" of the polynomial.

Adjacent ten polynomials are given where the polynomial degree is either

$m = 2$ (red font),
$m = 3$ (blue font) or
$m = 4$ (green font).

A polynomial $a(x)$ is called reducible if it can be represented as the product of two polynomials $p(x)$ and $q(x)$ of lower degree:

$$a(x) = p(x) \cdot q(x)$$

If this splitting is not possible, that is, if for the polynomial

$$a(x) = p(x) \cdot q(x) + r(x)$$

with a residual polynomial $r(x) ≠ 0$ holds, then the polynomial is called irreducible. Such irreducible polynomials are of special importance for the description of error correction methods.

⇒ The proof that a polynomial $a(x)$ of degree $m$ is irreducible requires several polynomial divisions $a(x)/q(x)$, where the degree of the respective divisor polynomial $q(x)$ is always smaller than $m$. Only if all these divisions $($modulo $2)$ always yield a remainder $r(x) ≠ 0$ it is proved that $a(x)$ is indeed an irreducible polynomial.

⇒ This exact proof is very complex. Necessary conditions for $a(x)$ to be an irreducible polynomial at all are the two conditions $($in nonbinary approach "$x=1$" would have to be replaced by "$x≠0$"$)$:

$a(x = 0) = 1$,

$a(x = 1) = 1$.

Otherwise, one could write for the polynomial under investigation:

$$a(x) = q(x) \cdot x \hspace{0.5cm}{\rm resp.}\hspace{0.5cm}a(x) = q(x) \cdot (x+1)\hspace{0.05cm}.$$

The above conditions are necessary, but not sufficient, as the following example shows:

$$a(x) = x^5 + x^4 +1 \hspace{0.3cm} \Rightarrow\hspace{0.3cm}a(x = 0) = 1\hspace{0.05cm},\hspace{0.2cm}a(x = 1) = 1 \hspace{0.05cm}.$$

Nevertheless, this polynomial is reducible:

$$a(x) = (x^3 + x +1)(x^2 + x +1) \hspace{0.05cm}.$$

Hint: This exercise belongs to the chapter "Extension field".

Questions

Solution

(1) The polynomial $a(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + \hspace{0.1cm}... \hspace{0.1cm} + a_m \cdot x^{m}$

with $a_m = 1$

and given coefficients $a_0, \ a_1, \hspace{0.05cm}\text{ ...} \hspace{0.1cm} , \ a_{m-1}$ $(0$ or $1$ in each case$)$

is irreducible if there is no single polynomial $q(x)$ such that the modulo $2$ division $a(x)/q(x)$ yields no remainder. The degree of all divisor polynomials $q(x)$ to be considered is at least $1$ and at most $m-1$.

For $m = 2$ two polynomial divisions $a(x)/q_i(x)$ are necessary, namely with

$$q_1(x) = x \hspace{0.5cm}{\rm and}\hspace{0.5cm}q_2(x) = x+1 \hspace{0.3cm} \Rightarrow\hspace{0.3cm}N_{\rm D}\hspace{0.15cm}\underline{= 2} \hspace{0.05cm}.$$

For $m = 3$ there are already $N_{\rm D} \ \underline{= 6}$ divisor polynomials, namely besides $q_1(x) = x$ and $q_2(x) = x + 1$ also

$$q_3(x) = x^2\hspace{0.05cm},\hspace{0.2cm}q_4(x) = x^2+1\hspace{0.05cm},\hspace{0.2cm} q_5(x) = x^2 + x\hspace{0.05cm},\hspace{0.2cm}q_6(x) = x^2+x+1\hspace{0.05cm}.$$

For $m = 4$, besides $q_1(x), \hspace{0.05cm}\text{ ...} \hspace{0.1cm} , \ q_6(x)$ all possible divisor polynomials with degree $m = 3$ must also be considered, thus:

$$q_i(x) = a_0 + a_1 \cdot x + a_2 \cdot x^2 + x^3\hspace{0.05cm},\hspace{0.5cm}a_0, a_1, a_2 \in \{0,1\} \hspace{0.05cm}.$$

For the index, $7 ≤ i ≤ 14 \ \Rightarrow \ N_{\rm D} \ \underline{= 14}$.

(2) For the first polynomial holds: $a_1(x = 0) = 0$. Therefore this polynomial is reducible:

$$a_1(x) = x \cdot (x + 1).$$

On the other hand, the following is true for the second polynomial:

$$a_2(x = 0) = 1\hspace{0.05cm},\hspace{0.2cm}a_2(x = 1) = 1 \hspace{0.05cm}.$$

This necessary but not sufficient property shows that $a_2(x)$ could be irreducible. The final proof is obtained only by two modulo-2 divisions:

$a_2(x)$ divided by $x$ yields $x + 1$, remainder $r(x) = 1$,

$a_2(x)$ divided by $x + 1$ yields $x$, remainder $r(x) = 1$.

The correct solution is therefore the proposed solution 2.

(3) The first three polynomials are reducible, as the following calculation results show:

$$a_3(x = 0) = 0\hspace{0.05cm},\hspace{0.2cm}a_4(x = 1) = 0\hspace{0.05cm},\hspace{0.2cm}a_5(x = 0) = 0\hspace{0.05cm},\hspace{0.2cm}a_5(x = 1) = 0 \hspace{0.05cm}.$$

This could have been found out by thinking:

$$a_3(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} x \cdot x \cdot x \hspace{0.05cm},$$

$$a_4(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (x^2 + x + 1)\cdot(x + 1) \hspace{0.05cm},$$

$$a_5(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} x \cdot (x + 1)\cdot(x + 1) \hspace{0.05cm}.$$

The polynomial $a_6(x)$ is not equal to $0$ for both $x = 0$ and $x = 1$ . This means that

"nothing speaks against" $a_6(x)$ being irreducible,
division by the irreducible degree-1 polynomials $x$ and $x + 1$, respectively, is not possible without remainder.

However, since division by the single irreducible degree-2 polynomial also yields a remainder, it is proved that $a_6(x)$ is an irreducible polynomial:

$$(x^3 + x+1)/(x^2 + x+1) = x + 1 \hspace{0.05cm},\hspace{0.4cm}{\rm Rest}\hspace{0.15cm} r(x) = x\hspace{0.05cm},$$

Using the same computational procedure, it can also be shown that $a_7(x)$ is also irreducible ⇒ Solutions 4 and 5.

(4) From $a_8(x + 1) = 0$ follows the reducibility of $a_8(x)$. For the other two polynomials, however, holds:

$$a_9(x = 0) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 1\hspace{0.05cm},\hspace{0.35cm}a_9(x = 1) = 1 \hspace{0.05cm},$$

$$a_{10}(x = 0) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}1\hspace{0.05cm},\hspace{0.2cm}a_{10}(x = 1) = 1 \hspace{0.05cm}.$$

So both could be irreducible. The exact proof of irreducibility is more complicated:

It is not necessary to use all four divisor polynomials with degree $m = 2$, namely $x^2, \ x^2 + 1, \ x^2 + x + 1$, but it is sufficient to divide by the only irreducible degree-2 polynomial. One obtains with respect to the polynomial $a_9(x)$:

$$(x^4 + x^3+1)/(x^2 + x+1) = x^2 + 1 \hspace{0.05cm},\hspace{0.4cm}{\rm remainder}\hspace{0.15cm} r(x) = x\hspace{0.05cm}.$$

Also the division by the two irreducible degree-3 polynomials yields a remainder in each case:

$$(x^4 + x^3+1)/(x^3 + x+1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} x + 1 \hspace{0.05cm},\hspace{0.4cm}{\rm remainder}\hspace{0.15cm} r(x) = x^2\hspace{0.05cm},$$

$$(x^4 + x^3+1)/(x^3 + x^2+1) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} x \hspace{0.05cm},\hspace{0.95cm}{\rm remainder}\hspace{0.15cm} r(x) = x +1\hspace{0.05cm}.$$

Finally, let us consider the polynomial $a_{10}(x) = x^4 + x^2 + 1$. Here holds

$$(x^4 + x^2+1)/(x^2 + x+1) = x^2 + x+1 \hspace{0.05cm},\hspace{0.4cm}{\rm remainder}\hspace{0.15cm} r(x) = 0\hspace{0.05cm}.$$

It follows: Only the polynomial $a_9(x)$ is irreducible ⇒ proposed solution 2.

	$a_3(x) = x^3$,
	$a_4(x) = x^3 + 1$,
	$a_5(x) = x^3 + x$,
	$a_6(x) = x^3 + x + 1$,
	$a_7(x) = x^3 + x^2 + 1$.

	$a_8(x) = x^4 + 1$,
	$a_9(x) = x^4 + x^3 + 1$,
	$a_{10}(x) = x^4 + x^2 + 1$.