Question

Let p (x)be irreducible in F [X]. Without using Theorem 5.10, prove that if $\left[f\left(x\right)\right]\left[g\left(x\right)\right]\mathbf{=}\left[0_F\right]$ in F [x]/(p (x)) then $\left[f\left(x\right)\right]\mathbf{=}\left[0_F\right]\mathbf{}\mathit{o}\mathit{r}\mathbf{}\left[g\left(x\right)\right]\mathbf{=}\left[0_F\right]$. [Hint: Exercise 10 in section 5.1.]

Short answer

It is proved $\left[\mathrm{f}\left(\mathrm{x}\right)\right]=\left[0_{\mathrm{F}}\right]and\left[\mathrm{g}\left(\mathrm{x}\right)\right]=\left[0_{\mathrm{F}}\right]$.

Step-by-step solution

Statement of Exercise 5.1.10

Exercise 5.1.10states when p (x) is irreducible in F [x] and $f\left(x\right)g\left(x\right)\equiv 0_F\left(\mathrm{mod}p\left(x\right)\right)$, then $f\left(x\right)\equiv 0_F\left(\mathrm{mod}p\left(x\right)\right)$ and $g\left(x\right)\equiv 0_F\left(\mathrm{mod}p\left(x\right)\right)$ .

Show that when [fx][gx]=[0F] in F [x]/ (p (x)) , then [f(x)]=[0F] and  [g(x)]=[0F]

When $\left[\mathbf{f}\left(\mathbf{x}\right)\right]\left[\mathbf{g}\left(\mathbf{x}\right)\right]=\left[0_F\right]$in F [x]/ (p (x)), then it may deduce that $f\left(x\right)g\left(x\right)\equiv 0_F\left(\mathrm{mod}p\left(x\right)\right)$ from unpacking the definitions. According to exercise 5.1.10, either $f\left(x\right)\equiv 0\left(\mathrm{mod}p\left(x\right)\right)$ or $g\left(x\right)\equiv 0\left(\mathrm{mod}p\left(x\right)\right)$. These last statements are equal to $\left[f\left(x\right)\right]=\left[0\right]$ and $\left[g\left(x\right)\right]=\left[0\right]$ .

Hence, it is proved$\left[\mathrm{f}\left(\mathrm{x}\right)\right]=\left[0_{\mathrm{F}}\right]and\left[\mathrm{g}\left(\mathrm{x}\right)\right]=\left[0_{\mathrm{F}}\right]$.