Given a polynomial ring $F[x_1…x_n]$ over a field $F$, I wanted to ask about when we can factor some $f \in F[x_1…x_n].$ I was curious to see whether a fundamental theorem of algebra-esque notion exists for every polynomial ring (i.e., we can always devise a field extension wherein all polynomials of our given field can be factored).
My thought process is inspired by the case of $\mathbb{Q}= F$ – here, we cannot factor polynomials such as $f(x) = x^2 – 2$, so we devise the algebraic extension $\mathbb{Q}(x)/x^2-2$, quotienting by the ideal generated by $f$.
Similarly, it seems that for an arbitrary case, we can simply consider $\mathbb{F}[x_1…x_n]/f$, which is a field whenever $f$ is irreducible. Therefore, perhaps we get the restriction that we can always construct field extensions wherein all irreducible polynomials are factorable. This is Idea 1.
However, Idea 1 is the same as Idea 2, wherein we take some wonderful object $a$ such that it satisfies the relation $f(a) = a^2 – 2=0$, adjoin it to $\mathbb{Q}$, and yield $\mathbb{Q}(a)$ – a field wherein $f$ as above now factors. We know that Idea 1 is isomorphic to Idea 2, but somehow in Idea 2, we bring out a strange element and (assuming $f$ is irreducible), derive a field by adjoining it to $\mathbb{Q}$.
Now, I imagine that when we have $f$ reducible, then we cannot do this if the polynomial ring is not a UFD – but in that case, what stops us from taking our field $F$, and simply adjoining it to it objects $a_i$ defined by the relation that $f(a_i) = 0$? I guess it would not be a field any longer, but what exactly breaks down? Moreover, are there examples of fields which don't admit any extension over which every polynomial splits? What do such examples tell us about our ability to devise some object, call it a root, and attach it to our field?
Best Answer
I want to add the following to Gerry's nice answer.
Several things break down when you have at least two variables. The questions becoming geometrical in nature, and hopefully a tag expert on algebraic-geometry can find a suitable older thread.