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) It is given that p and q are irreducible.. Let ud consider that p/q. then we can write it as q=up. Now we need to show that u is a unit.

Since q is irreducible then it a product of two polynomials out of which one has to be a unit. But p is also irreducible hence can neither be zero nor a unit. Therefore, u has to be a unit.

Conversely, if p and q are irreducible and q=up with u as a unit. Clearly, p divides q as p is one of the factors of q.

2) In addition if p and q are monic that is if they have degree=1. and p/q that is, q=up. If u is a non-constant then the product up gives a polynomial of degree>1 which voilates that q is a monic polynomial. Hence u is a constant , u=1 and this implies p=q