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12E

Expert-verified
Found in: Page 471

### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# Question 12: If is a generator polynomial of a BCH code in ${\mathrm{ℤ}}_{2}\left[x\right]}{\left({x}^{n}-1\right)}$, prove that divides $\left({x}^{n}-1\right)$. [Hint: Exercises 11 and 8(b).]

It has been proved that g(x) divides${x}^{n}-1$ .

See the step by step solution

## Step 1: Consider the results of

Exercise 11: Let $K={Z}_{2}\left(\alpha \right)$be a finite field of order ${2}^{r}$, whose multiplicative group is generated by . For each i, let${m}_{i}\left(x\right)$ be the polynomial of over . If ,${\mathbf{n}}{\mathbf{=}}{{\mathbf{2}}}^{{\mathbf{r}}}{\mathbf{-}}{\mathbf{1}}$ then each${m}_{i}\left(x\right)$ divides .

Exercise 8: If${{\mathbf{m}}}_{{\mathbf{1}}}\left(\mathbf{x}\right){\mathbf{,}}{{\mathbf{m}}}_{{\mathbf{2}}}\left(\mathbf{x}\right){\mathbf{,}}{.}{.}{.}{\mathbf{,}}{{\mathbf{m}}}_{{\mathbf{k}}}\left(\mathbf{x}\right)$ are distinct monic irreducible in$F\left[x\right]$ such that each${m}_{i}\left(x\right)$ divides $f\left(x\right)$, then $\mathrm{g}\left(\mathrm{x}\right):{\mathrm{m}}_{1}\left(\mathrm{x}\right),{\mathrm{m}}_{2}\left(\mathrm{x}\right),...,{\mathrm{m}}_{\mathrm{k}}\left(\mathrm{x}\right)$divides $f\left(x\right)$

## Step 2: Write the given data

Given that g(x)is a generator polynomial of a BCH code in ${\mathrm{ℤ}}_{2}\left[x\right]}{\left({x}^{n}-1\right)}$.

## Step 3: Determine the Proof

Here,$g\left(x\right)$ is a product of distinct minimal polynomials${m}_{i}\left(x\right)$ .

According to Exercise 11, ${m}_{i}\left(x\right)$divides ${x}_{n}-1$

This implies$g\left(x\right)$ divides ${x}_{n}-1$.

Hence, the result.