Chapter 8.6, Problem 10E

### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

Chapter
Section

### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
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:

43â€‰â€‰âˆ’â€‰â€‰223â€‰â€‰âˆ’â€‰â€‰2 in â„šâ€‰(â€‰23â€‰), where â„š is the field of rational numbers.

Explanation:

Let Î±â€‰â€‰=â€‰â€‰23â€‰â€‰

Taking cube from both sides, we get

Î±3â€‰â€‰=â€‰â€‰2â€‰â€‰

So p(Î±)â€‰â€‰=â€‰â€‰Î±3âˆ’â€‰â€‰2

Hence, p(x)â€‰â€‰=â€‰â€‰x3âˆ’â€‰â€‰2 is a polynomial which is irreducible over the field of rational number â„š by Eisenstein criteria.

Let 43â€‰â€‰âˆ’â€‰â€‰223â€‰â€‰âˆ’â€‰â€‰2 be written as (23)2â€‰âˆ’â€‰â€‰223â€‰â€‰âˆ’â€‰â€‰2.

Here, Î±â€‰â€‰=â€‰â€‰23â€‰ implies that fâ€‰(â€‰Î±â€‰)â€‰=â€‰Î±2â€‰âˆ’â€‰â€‰2Î±â€‰â€‰âˆ’â€‰â€‰2.

Hence, fâ€‰(â€‰xâ€‰)â€‰=â€‰x2â€‰âˆ’â€‰â€‰2xâ€‰â€‰âˆ’â€‰â€‰2

When pâ€‰(â€‰xâ€‰) is divided by fâ€‰(â€‰xâ€‰), we get

pâ€‰(â€‰xâ€‰)â€‰â€‰=â€‰â€‰f(â€‰xâ€‰)(â€‰xâ€‰+â€‰2â€‰)â€‰â€‰+â€‰â€‰6xâ€‰â€‰+â€‰â€‰2

This implies that â€‰6xâ€‰â€‰+â€‰â€‰2â€‰â€‰=â€‰pâ€‰(â€‰xâ€‰)â€‰â€‰âˆ’â€‰f(â€‰xâ€‰)(â€‰xâ€‰+â€‰2â€‰)

When fâ€‰(â€‰xâ€‰) is divided by 6xâ€‰â€‰+â€‰â€‰2, we get

f(x)â€‰â€‰=â€‰â€‰(â€‰6x+2â€‰)(â€‰16â€‰xâ€‰â€‰âˆ’â€‰â€‰718â€‰)â€‰â€‰âˆ’â€‰â€‰119

âˆ’119â€‰â€‰â€‰=â€‰â€‰fâ€‰(â€‰xâ€‰)â€‰â€‰âˆ’â€‰â€‰(â€‰6x+2â€‰)(â€‰16â€‰xâ€‰â€‰âˆ’â€‰â€‰718â€‰)

Hence, 1 is the greatest common divisor of pâ€‰(â€‰xâ€‰) and fâ€‰(â€‰xâ€‰)

1â€‰â€‰=â€‰â€‰(âˆ’911)(âˆ’119)â€‰

1â€‰â€‰=â€‰â€‰(âˆ’911)[fâ€‰(â€‰xâ€‰)â€‰â€‰âˆ’â€‰â€‰(â€‰6x+2â€‰)(â€‰16â€‰xâ€‰â€‰âˆ’â€‰â€‰718â€‰)]

1â€‰â€‰=â€‰â€‰(âˆ’911)[fâ€‰(â€‰xâ€‰)â€‰â€‰âˆ’â€‰â€‰[pâ€‰(â€‰xâ€‰)â€‰â€‰âˆ’â€‰f(â€‰xâ€‰)(â€‰xâ€‰+â€‰2â€‰)](â€‰16â€‰xâ€‰â€‰âˆ’â€‰â€‰718â€‰)]

1â€‰â€‰=â€‰â€‰(âˆ’911)[fâ€‰(â€‰xâ€‰)â€‰â€‰+â€‰â€‰[f(â€‰xâ€‰)(â€‰xâ€‰+â€‰2â€‰)â€‰â€‰âˆ’â€‰pâ€‰(â€‰xâ€‰)](â€‰16â€‰xâ€‰â€‰âˆ’â€‰â€‰718â€‰)]

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