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12E

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Abstract Algebra: An Introduction

Found in: Page 471

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

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

Answer

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the results of

Exercise 11: Let $K = Z_{2} (α)$ be a finite field of order $2^{r}$ , whose multiplicative group is generated by . For each i, let $m_{i} (x)$ be the polynomial of over . If , $n = 2^{r} - 1$ then each $m_{i} (x)$ divides .

Exercise 8: If $m_{1} (x), m_{2} (x), . . ., m_{k} (x)$ are distinct monic irreducible in $F [x]$ such that each $m_{i} (x)$ divides $f (x)$ , then $g (x) : m_{1} (x), m_{2} (x), . . ., m_{k} (x)$ divides $f (x)$

Step 2: Write the given data

Given that g(x)is a generator polynomial of a BCH code in $\frac{ℤ_{2} [x]}{(x^{n} - 1)}$ .

Step 3: Determine the Proof

Here, $g (x)$ is a product of distinct minimal polynomials $m_{i} (x)$ .

According to Exercise 11, $m_{i} (x)$ divides $x_{n} - 1$

This implies $g (x)$ divides $x_{n} - 1$ .

Hence, the result.

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