Let R be a domain and let px qx Rx If p and q are irreducibl

Let R be a domain, and let p(x). q(x) R[x], If p and q are irreducible, prove that p | q if and only if there is a unit u with q =up. If, in addition, both p and q are monic, prove that p | q implies u = 1 and p = q.

Solution

1) Let p and q be irreducible.

As p/q there fore, q=up. As q is irreducible, then either of u or p has to be a unit. But p is already given to be irreducible therefore, p can neither be zero nor a unit. Hence we can say that u is a unit.

Conversely, if u is a unit such that q=up and p and q are irreducible hence neither of them is zero and it proves p/q.

2) in addition, if p and q are monic that is, they are of degree1. if p/q implies q=up. We can say that u is a constant because any degree of u greter than or equal to 1 can contradict that q is monic. Hence u=1 which in turn implies that p=q