Question on the Irreducibility of Polynomials

[Math Amateur]

I am reading Dummit and Foote on Polynomial Rings. In particular I am seeking to understand Section 9.4 on Irreducibility Criteria.

Proposition 9 in Section 9.4 reads as follows:

Proposition 9. Let F be a field and let $$p(x) \in F[x]$$. Then p(x) has a factor of degree one if and only if p(x) has a root in F i.e. there is an $$\alpha \in F$$ with $$p( \alpha ) = 0$$

Then D&F state that Proposition 9 gives a criterion for irreducibility for polynomials of small degree

D&F then state Proposition 10 as follows:

Proposition 10: A polynomial of degree two or three over a field F is reducible if and only if it has a root in FBUT! Here is my problem - why does not a root in F imply reducibility in polynomials of all degrees? A root in F means, I think, that the polynomial concerned has a linear factor and hence can be factored into a linear factor times a polynomial of degree n-1?

Can anyone clarify this for me?

Peter

[This question has also been posted on MHF]

 

[Fernando Revilla]

BUT! Here is my problem - why does not a root in F imply reducibility in polynomials of all degrees? A root in F means, I think, that the polynomial concerned has a linear factor and hence can be factored into a linear factor times a polynomial of degree n-1?

You are right, but take into account that proposition 10 says if and only if. For example consider $F=\mathbb{R}$ and $p(x)=(x^2+1)(x^2+2)$. This polynomial is obviously reducible in $\mathbb{R} [x]$ and has no real roots.