An interesting mathoverflow question was one due to Philipp Lampe that asked whether a non-surjective polynomial function on an infinite field can miss only finitely many values. In my interpretation of the question, if $k$ is a starting field and $f$ is a polynomial, you could ask what happens if you repeatedly adjoin a root of $f(x)-a$, except for a finite set of values $a in S subset k$ for which you hope a root never appears. You have to adjoin a root for all $a in tilde{k} setminus S$, where $tilde{k}$ is the growing field. Either a root of $f(x)-a$ for some $a in S$ will eventually appear by accident, or $f$ as a polynomial over the limiting field $tilde{k}$ is an example.
(Edit: I call this an interpretation rather than a construction, because in generality it is equivalent to Philipp's original question. I also don't mean to claim credit for the idea; it was already under discussion when I posted my answer then. Maybe an answer to the question below was already implied in the previous discussion, but if so, I didn't follow it.)
For some choices of $f$ and a non-value $a$, you can know that you are sunk at the first stage. For instance, suppose that $f(x) = x^n$. When you adjoin a root of $x^n-a$, you also adjoin a root of $x^n-b^na$ for every $b in k$. You cannot miss $a$ without also missing every $b^na$, which is then infinitely many values when $k$ is infinite.
So let $k$ be an infinite field, and let $f in k[x]$ be a polynomial. Define an equivalence relation on those elements $a in k$ such that $f(x)-a$ is irreducible. The relation is that $a sim b$ if adjoining one root of $f(x)-a$ and $f(x)-b$ yield isomorphic field extensions of $k$. Is any such equivalence class finite? What if $k$ is $mathbb{Q}$ or a number field?
In my partial answer to the original MO question, I calculated that if $f$ is cubic and the characteristic of $k$ is not 2 or 3, then the equivalence classes are all infinite.
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