Question

Write out a complete proof of Theorem 5.6 (that is, carry over to F [x] the

proof of the analogous facts for Z).

Short answer

The proof is completed of theorem 5.6.

Step-by-step solution

Determine theorem 5.6

$\left[f\left(x\right)\right]=\left[g\left(x\right)\right]$and $\left[h\left(x\right)\right]=\left[k\left(x\right)\right]$

By using theorem 5.3,

$f\left(x\right)=g\left(x\right)\left(\mathrm{mod}p\left(x\right)\right)$and $h\left(x\right)\equiv k\left(x\right)\left(\mathrm{mod}p\left(x\right)\right)$; thus, using theorem 5.2

$f\left(x\right)+h\left(x\right)\equiv g\left(x\right)+k\left(x\right)\left(\mathrm{mod}p\left(x\right)\right)$and $f\left(x\right)h\left(x\right)=g\left(x\right)k\left(x\right)\left(\mathrm{mod}p\left(x\right)\right)$

Hence, again apply theorem 5.3.

$\left[f\left(x\right)+h\left(x\right)\right]=\left[g\left(x\right)+k\left(x\right)\right]$and $\left[f\left(x\right)h\left(x\right)\right]=\left[g\left(x\right)k\left(x\right)\right]$.