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I'm stuck on the following problem from a notes.

Let $B=\{(a_0,...,a_{n-1})\in\mathbb{C}^n:P_a(x)=x^n+a_{n-1}x^{n-1}+...+a_0\ \text{ is square free}\}\ $ and $E=\{(x,a)\in\mathbb{C}\times B:P_a(x)=0\}$ Prove that:$p:E\longrightarrow B,\ p(x,a)=a$ is a covering map and is not trivializable unless $n=1$

My attempts: I trying to show that $E$ is connected,then by another question If a covering map has a section, is it a $1$-fold cover?,we can solve the problem.But I don't know how to show $E$ is connected.

My question: Can we solve the problem by showing that $E$ is connected? And if not,how can I solve the original problem?

Any help is appreciated and thanks in advance!

Kevin.S
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Rixinner
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  • How would you construct a section? – Paul Frost Apr 04 '22 at 13:50
  • @PaulFrost:We are asked to show that E is not trivializable,so if we assume that E is trivializable,we can easily get a section. – Rixinner Apr 05 '22 at 13:26
  • That coverings with a section are 1-fold is only true if $E$ is connected. – Paul Frost Apr 05 '22 at 22:56
  • @PaulFrost:Yes,so I say above that I'm tring to show E is connected. – Rixinner Apr 07 '22 at 07:55
  • But this approach cannot work. The fibers $p^{-1}(a)$ have $n$ elements, thus there cannot be a section (unless $n = 1$). – Paul Frost Apr 07 '22 at 08:13
  • @PaulFrost:Thank you for such a quick reply.But I don't quite understand you.Can you be more specific?I'm arguing by contradiction.Assuming that E is trivializable,there must be a section.Then if we show that $E$ is connected,we can get $E$ must be a 1-fold covering,a contradiction unless $n=1$. – Rixinner Apr 07 '22 at 08:49
  • I think I was mislead by the "My question" part. The essence is to show that $p$ is a covering map. Only if we know this, a positive answer to your question "Can we solve the problem by showing that $E$ is connected?" correctly shows that $p$ is not trivializable. Thus your question only addresses a part of the problem. – Paul Frost Apr 07 '22 at 10:03
  • @PaulFrost:I'm sorry for misleading you,I will edit my question. – Rixinner Apr 08 '22 at 03:42

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