Let denote the set of polynomials with rational coefficients and integer constant terms.
(a) If p is prime in , prove that the constant polynomial p is irreducible in .
(b) If p and q are positive primes in , prove that p and q are not associates in
I am unsure of my thinking on these problems.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------
Regarding (a) I think the solution is as follows:
We need to show the p is irreducible in
That is if p = ab for then at least one of a or b must be a unit
But then we must have p = 1.p = p.1 since p is a prime in and hence is prime in
But 1 is a unit in (and also in )
Thus p is irreducible in
---------------------------------------------------------------------------------------------------------------------------------------------------------------------
Could someone please either confirm that my working is correct in (a) or let me know if my reasoning is incorrect or lacking in rigour.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Help with the general approach for (b) would be appreciated
Peter