Can someone verify whether or not this proof works for me? Thanks.
Suppose that and is a finite extension of . If is irreducible over and and are relatively prime, show that is irreducible over .
Suppose is reducible over . Then has a root in , call it (this part feels true but I can't verify it nicely...basically if it's reducible but not by splitting out , the coefficients will be from , but then it would be reducible over ). Take the subextension . Then since we have . Then but this implies that divides which it does not since and and are relatively prime.