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

Ans monic polynomial suffias to to prove that t[n] has element of k. The fact that K(a) is we first prove that sko KG) has infinitely many irreducible has infinitely manimal infinitely many ideals. Since K is a field, KG] is a principal ideal domain (PID) and the uncle of kn) are precisely the P.I.D means any ideal Isk(n) is maximal eff I =*f> for irreducible f. The fact that the units are the elements of k means inreducible polynomed f and g are g are anociates iff fag.

This f and irreducible Plynomials and fig implies vo monic that ef> and (g) are distinct masinal ideals of K[n) Case 1: k is a Enfinite. Then the set { (n-a)/ ack} is an infinite set of irreducible monic Polynomial and as above we are done. Case 2: k is finite. Kafa (say), is finite field then each a there is opgeba a splitting field of arn in which is fyn = K (1) (W) 7(A.Cu)) where for irredudicle of degreen so we have degree in' is reducible polynomial over k for every Possfive integer na