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Expert-verified Found in: Page 104 ### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624 # Express ${x}_{4}-4$ as a product of irreducible in $\mathrm{ℚ}\left[x\right]$, in role="math" localid="1648646593814" $\mathrm{ℝ}\left[x\right]$, and in $\mathrm{ℂ}\left[x\right]$.

The factorization in $\mathrm{ℚ}\left[x\right]$ is ${x}^{4}-4=\left({x}^{2}-2\right)\left({x}^{2}+2\right)$.

The factorization in $\mathrm{ℝ}\left[x\right]$ is role="math" localid="1648647843738" ${x}^{4}-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left({x}^{2}+2\right)$.

The factorization in role="math" localid="1648646692368" $\mathrm{ℂ}\left[x\right]$ is ${x}^{4}-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left(x+i\sqrt{2}\right)\left(x-i\sqrt{2}\right)$.

See the step by step solution

## Step 1: Determine x4-4

Consider the given function, ${x}^{4}-4$

The factorization in $\mathrm{ℚ}\left[x\right]$ is ${x}^{4}-4=\left({x}^{2}-2\right)\left({x}^{2}+2\right)$.

In $\mathrm{ℝ}\left[x\right],{x}^{2}-2-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)$, the factorization is,

${x}^{4}-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left({x}^{2}+2\right)$.

## Step 2: Determine  x4-4in ℂx

Now, in $\mathrm{ℂ}\left[x\right]$, ${x}^{2}+2-\left(x+i\sqrt{2}\right)\left(x-i\sqrt{2}\right)$ the factorization is,

${x}^{4}-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left(x+i\sqrt{2}\right)\left(x-i\sqrt{2}\right)$. ### Want to see more solutions like these? 