Question

Suppose p is an irreducible element in an integral domain Rsuch that whenever $\mathbf{p}\mathbf{|}\mathbf{bc}$, then $\mathbf{p}\mathbf{|}\mathbf{b}$or $\mathbf{p}\mathbf{|}\mathbf{c}$. If $\mathbf{p}\mathbf{|}{\mathbf{a}}_{\mathbf{1}}{\mathbf{a}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{a}}_{\mathbf{n}}$, prove that p divides at least one a_i.

Short answer

It is proved that p divides at least onea_i.

Step-by-step solution

Referring to the Theorem 10.15

Theorem 10.15

Let P be an irreducible element in a unique factorization domain R. If $\mathbf{p}\mathbf{|}\mathbf{bc}$then $\mathbf{p}\mathbf{|}\mathbf{b}$or $\mathbf{p}\mathbf{|}\mathbf{c}$.

Given that p is an irreducible element in an integral domain R.

Proving that p divides at least one ai 

As given in the question, whenever $\mathrm{p}|\mathrm{bc}$then $\mathrm{p}|\mathrm{b}$or $\mathrm{p}|\mathrm{c}$.

Therefore, byTheorem 10.15, it is a unique factorization domain R.

So, according to the theorem, if $\mathrm{p}|{\mathrm{a}}_1{\mathrm{a}}_2....{\mathrm{a}}_{\mathrm{n}}$then $\mathrm{p}|{\mathrm{a}}_1$or $\mathrm{p}|{\mathrm{a}}_2$....or$\mathrm{p}|{\mathrm{a}}_{\mathrm{n}}$

Therefore, for any $0\le \mathrm{i}\le \mathrm{n}$, we can conclude that p divides any one of a_i.

Hence, it is proved that p divides at least one a_i.