I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following:
Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if and only if $p(x)$ has a root in $F$.
Proposition 10: a polynomial of degree 2 or 3 over a field $F$ is reducible if and only if it has a root in $F$
My question: Why is this not necessarily true for polynomials degree $>3$? If the linear factor is $q(x)$ of some polynomial $f(x)$ degree 73, can't it still be written as $f(x)=q(x)f'(x)$ for $f'(x)$ of degree 72?
The problem is the "only if" part of the latter statement. Consider the polynomial $x^4 + 2x^2 + 1 = (x^2+1)^2 \in \Bbb R[x]$. It has no root in $\Bbb R$, but still it is reducible.
You are completely correct, though: The "if" part of the statement remains true. Any polynomial which has a root and degree greater than one is reducible over any field.