Weakly normal polynomials and normal polynomials.

Question

I have been going through the notes of Prof. Pete L. Clark here (warning: long pdf). They are rough notes on Field theory and on page 30 he defines $P \in K[t]$ a normal polynomial if $P \in L[t]$ is either irreducible or split for all $L$ extensions of $K$.

Further, $P \in K[t]$ is a weakly normal polynomial if it splits in $L/K$ whenever $P$ has a root in $L$. An exercise is to show that if $P$ is non split and weakly normal, then it is of the form $Q(t)^a$ for some $Q \in K[t]$.

However, wouldn't $P(t) = (t^2+3)(t^2 + t +1)$ give a counterexample to the proposition whenever $K$ has characteristic $\neq 2$ and $3$ is not a square? The $3$rd root of unity $\omega = (1 + \sqrt{-3})/2$ and therefore both the polynomials dividing $P(t)$ split together.

Answer 1

Here are the definitions together with the full statement of the exercise: $\newcommand{\Z}{\mathbb{Z}}$

A nonconstant polynomial $P \in K[t]$ is a normal polynomial if for all field extensions $L/K$, $P \in L[t]$ is either irreducible or split. A nonconstant polynomial $P \in K[t]$ is weakly normal if for every field extension $L/K$, if $P$ has a root in $L$ then $P \in L[t]$ is split.

Exercise: Let $P \in K[t]$ be a polynomial of degree at least $2$.
a) Show that if $P$ is normal, it is weakly normal.
b) Suppose $P$ is nonsplit and weakly normal. Show that there is a nonsplit normal polynomial $Q \in K[t]$ and $a \in \Z^+$ such that $P(t) = Q(t)^a$.
c) Deduce that an irreducible weakly normal polynomial is normal.
d) Deduce that a weakly normal polynomial of prime degree is normal.

I agree that you have given a counterexample to part b) of this exercise. This makes me wonder about parts c) and d)!

This material on normal polynomials was, at best, half-baked. The only result in the section is a proposition, saying that for a polynomial $P \in K[T]$ of degree at least $2$, $P$ is nonsplit normal iff $K[T]/(P)$ is a normal field extension of $K$. But the proof of that result is incomplete.

I believe I was taking this material from some older source on field theory, but I never found the time to work it through. I have simply removed this section from the notes, but when I get a chance, maybe I can include it back in the future.