Question

Prove that the minimal polynomial of ${\mathbf{\alpha }}^{\mathbf{5}}$over${\mathbf{Z}}_{\mathbf{2}}$ is$\mathbf{1}\mathbf{+}\mathbf{x}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}$ [Hint use the table in example1]

Short answer

It is proved that the minimal polynomial of ${\alpha }^5$over$Z_2$ is obtained as: $1+x+x^2$

Step-by-step solution

Conceptual Introduction

Coding theory is the study of a code's characteristics and how well-suited it is to various applications. Data compression, cryptography, error detection, and correction, as well as data transport and storage, all depend on codes.

Determine the proof for the polynomial:

Let $\alpha$represent the primitive element of $Z_2\left(x\right)$ so that $1+\alpha +{\alpha }^4=0$.The minimum polynomials $\alpha ,{\alpha }^3,{\alpha }^5$of are:

$$\begin{gathered} m_1\left(x\right)=1+x+x^4 \\ {m}_3\left(x\right)=1+x+x^2+x^3+x^4 \\ {m}_5\left(x\right)=1+x+x^2 \end{gathered}$$

The powers of polynomials are calculated by using the factor $1+\alpha +{\alpha }^4=0$as $k$

Represents the root of the polynomial $\alpha$in $1+x+x^2$

Consider the reference for the table in example 1

Exponential Notation	Polynomial Notation	Binary Notation	Decimal Notation	Minimal Notation
${\alpha }^5$	$Z+Z^2$	$0110$	$6$	$1+x+x^2$

Hence, the minimal polynomial of ${\alpha }^5$ over $Z_2$ is $1+x+x^2$ .