Question

Factor each polynomial as a product of irreducible polynomials in $\mathbf{\square }\left[x\right]$, in $\mathbf{\square }\mathbf{\left[}\mathbf{x}\mathbf{\right]}$, and in $\mathbf{\square }\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

$\left(a\right){\mathit{x}}^{\mathbf{4}}\mathbf{‐}\mathbf{2}$

Short answer

The factors as a product of irreducible polynomials are:

$$\begin{gathered} \square \left[x\right]=x^4‐2 \\ \square \left[x\right]=\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right)\left(x^2+\sqrt{2}\right) \\ \square \left[x\right]=\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right)\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right) \end{gathered}$$

Step-by-step solution

Definition of Irreducible Polynomial.

A polynomial is called irreducible if it cannot be factorized into a non-trivial polynomial over the same field.

Finding the irreducible polynomial of x4‐2 .

The given polynomial is $x^4‐2$.

The irreducible polynomial of$x^4‐2$can be given as:

$$\begin{gathered} f\left(x\right)=x^4‐2 \\ Q\left[x\right]=x^4‐2 \\ R\left[x\right]=\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right)\left(x^2+\sqrt{2}\right) \\ C\left[x\right]=\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right)\left(x^2‐i\sqrt[4]{2}\right)\left(x^2+i\sqrt[4]{2}\right) \end{gathered}$$

Hence, the factors as a product of irreducible polynomials are:

$$\begin{gathered} \square \left[x\right]=x^4‐2 \\ \square \left[x\right]=\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right)\left(x^2+\sqrt{2}\right) \\ \square \left[x\right]=\left(x^2‐\sqrt[4]{2}\right)\left(x^2+\sqrt[4]{2}\right)\left(x^2‐i\sqrt[4]{2}\right)\left(x^2+i\sqrt[4]{2}\right) \end{gathered}$$