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Expert-verified Found in: Page 94 ### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624 # Show that 1+3x is a unit in ${\mathrm{ℤ}}_{9}\left[x\right]$ . Hence, Corollary 4.5 may be false if R is not an integral domain.

It is proved that 1+3x is a unit in ${\mathrm{ℤ}}_{9}\left[x\right]$ .

See the step by step solution

## Step 1: Polynomial Arithmetic

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

## Step 2: Fields:

The given unit is 1+3x . The multiplicative inverse of this will be: 1-3x.

Then, we have:

$\left(1+3x\right)\left(1-3x\right)=1-9{x}^{2}=1$

Clearly found to be in ${\mathrm{ℤ}}_{9}\left[x\right]$ .

Hence proved, 1+3x is a unit in ${\mathrm{ℤ}}_{9}\left[x\right]$ . ### Want to see more solutions like these? 