Alright. I have this three-part assignment, but im really stuck on the second question. So if you could give me some pointers, it would be nice.
Let (Where the F is the double-stroke F. Didnt know how to do this.)
(i) Show that are the only monic irreducible polynomials of degree 2 in R.
I have already solved this question. I just listed all monic polynomials of degree 2 in R, and made a table showing which functions had roots and which where irreducible.
(ii) Show that if a polynomial of degree 4 or 5 with no roots is reducible, then there is a monic irreducible polynomial of degree 2 dividing
This is the thing I dont know how to prove.
A section of my textbook says that a polynomial ring (in this case ) is a unique factorization domain, which means that if isn't irreducible, there is a factorization such that and .
I also have a part that says that if is a domain (which I believe is the case here), we have that where is defined as the set of roots of .
So we can say, that there exists and , where both have no roots.
I have a feeling, that's the way I'm supposed to go, but I can't quite figure out how to tie it all together. So if someone could give me a hint, or tell me if im going the wrong way?
Thanks in advance.