I want to restrict myself to polynomials with real-valued coefficients and consider only real-valued roots (so, in this context, for instance, a 2nd degree polynomials sometimes cannot be factorized).
In this context, can a 4th degree polynomial always be factorized in two 2nd-degree polynomials ?
Or is it possible to come up with a 4th degree polynomials which could not be factorized into any real-coefficients polynomials ?
(and - this maybe should be the object of a further question - but could we extend this property to all even degree polynomials ?)
The ring of one-variable polynomials over any field is a Euclidean ring (it means that the divisibility properties in it are somewhat similar to the ones in $\mathbb{Z}$). Particularly, that means, that all polynomials can be factorized in a product of "prime" polynomials (usually called "irreducible"), that can not be divided by any polynomial other then a non-zero constant or themselves multiplied by a non-zero constant (non-zero constant polynomials play here a role similar to the one of $\{1; -1\}$ in $\mathbb{Z}$). The irreducible polynomials over $\mathbb{R}$ are well classifies. There are two classes of them:
1)$ax + b$
2)$ax^2 + bx + c$, where $b^2 < 4ac$
Thus there are four options for our fourth degree polynomial. It is either $(ax^2 + bx + c)(dx^2 + ex + f)$ (with $b^2 < 4ac$ and $e^2 < 4df$) or $(ax^2 + bx + c)(dx + e)(fx + g)$ (with $b^2 < 4ac$) or $(ax+b)(cx+d)(ex + f)(gx + h)$.