Question

Show that if K is a field, then K[2] contains infinitely many irreducibles. (Note: keep in mind that K could be finite.)

Answer 1

means are Proof: We first showing ne lenew to that We first prove that showing that Kea] has infinitely many irreducible monic polynomial Suffices to prove that keaI has infinitely maximal ideals. many Since K. is a field, k[a] is a poincipal ideal domain On (PID) and the units of k[a] are precisely the element of ki The fact that k[a] is P.J.P means ad ideal amy I Çk[a] is maximal iff. I = Lf) for irreducible of The fact that the units are the element of k irreducible polynomial of and 8 ansociales off fag. Thus f and monic polynomials and of ag implies egg are distinct manimal ideals of keng kis & infinite the set & Cara) I ack} is an infinite set of irreducible monic polynomial and as above we are done. Case-ll kis finite K= F (909), is a froute field then each in there splifting field of gth x Fin & KER] / (fin (as) , there for is irreducible of degree degree irreducible polynomial over for every positive infeger no irreducible that (f) and Case-l Then is a cich is so we have ni