Showing Difference of Relatively Prime Polynomials is Irreducible

  • #1

Homework Statement


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].


Homework Equations


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.
 

Answers and Replies

  • #2
354
0
...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]),
Correct.
but the left hand side is most certainly in the original ring.
This is incorrect. Each part of the equation [itex]hk(af+bg)=hk=f-yg[/itex] was derived by directly applying noncontradictory definitions (namely (i) [itex] af+bg := 1[/itex] and (ii) [itex]hk := f-yg[/itex]), so you won't be able to get a contradiction without doing something else.
 
  • #3
354
0
The problem is I don't feel confident at all in this argument. I am having trouble conceptualizing what [tex]f-yg[/tex] is.

Think of [itex]y[/itex] as a constant (which is what it is). It might help to use a different letter, say [itex]\alpha[/itex], instead of [itex]y[/itex] for the time being so you don't accidentally forget it's not a variable.
 

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