Let [tex]K[/tex] be a field, and [tex] f,g[/tex] are relatively prime in [tex] K[x][/tex]. Show that [tex]f-yg[/tex] is irreducible in [tex] K(y)[x][/tex].
There exist polynomials [tex]a,b\in K[x][/tex] such that [tex] af+bg=u[/tex] where [tex]u\in K[/tex]. We also have the Euclidean algorithm for polynomials.
The Attempt at a Solution
Assuming towards a contradiction that [tex] f-yg[/tex] were reducible, we have [tex]f-yg=hk[/tex] where [tex] h,k\in K(y)[x][/tex] are not units. Then by the relative primacy condition we also have [tex] af+bg=1[/tex], so that multiplying both sides by [tex]hk[/tex] yields [tex]hk(af+bg)=hk=f-yg[/tex], but this is a contradiction since [tex]f-yg[/tex] is certainly not in our original ring of polynomials (assuming that [tex] y\notin K[/tex]), but the left hand side is most certainly in the original ring. The problem is I don't feel confident at all in this argument. I am having trouble conceptualizing what [tex]f-yg[/tex] is.