Hello I am just in an intro to algebra course and I am having some problems completely understanding what is meant by irreducible polynomial.
Partially this is because I actually have two definitions in my notes, which I am sure are equivalent, but it is just making me confused.
One definition is that first we define the equivalence relation ~ on polynomials in a field by f(x)~g(x) iff $f(x)=ag(x)$ for some non zero $a$ in the field.
then we say f is irreducible if g|f implies that g ~ 1 or g~f.
And second is that we say a non constant polynomial is irreducible in a field $F[X]$ if $f=qs$ then either q or s are units.
I'm just not exactly sure I understand what is going on, like what it is really saying. I don't have any examples either, and am not sure which definition makes more sense etc. Hopefully someone can help with my understanding.