   Chapter 8.6, Problem 10E

Chapter
Section
Textbook Problem

# Find the multiplicative inverse of 4 3     −     2   2 3     −     2 in ℚ ( 2 3 ) , where ℚ is the field of rational numbers.

To determine

The multiplicative inverse of 432232 in (23), where is the field of rational numbers.

Explanation

Given information:

432232 in (23), where is the field of rational numbers.

Explanation:

Let α=23

Taking cube from both sides, we get

α3=2

So p(α)=α32

Hence, p(x)=x32 is a polynomial which is irreducible over the field of rational number by Eisenstein criteria.

Let 432232 be written as (23)22232.

Here, α=23 implies that f(α)=α22α2.

Hence, f(x)=x22x2

When p(x) is divided by f(x), we get

p(x)=f(x)(x+2)+6x+2

This implies that 6x+2=p(x)f(x)(x+2)

When f(x) is divided by 6x+2, we get

f(x)=(6x+2)(16x718)119

119=f(x)(6x+2)(16x718)

Hence, 1 is the greatest common divisor of p(x) and f(x)

1=(911)(119)

1=(911)[f(x)(6x+2)(16x718)]

1=(911)[f(x)[p(x)f(x)(x+2)](16x718)]

1=(911)[f(x)+[f(x)(x+2)p(x)](16x718)]

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