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.