[Math] Multiple roots of polynomials over a finite field

abstract-algebrafield-theorypolynomials

  1. Show that $x^4+x+1$ over $\mathbb{Z}_2$ does not have any multiple zeros in any field extension of $\mathbb{Z}_2$.
  2. Show that $x^{21} + 2x^8 +1$ does not have multiple zeros in any extension of $\mathbb{Z}_3$.
  3. Show that $x^{21} + 2x^9 +1$ has multiple zeros in some extension of $\mathbb{Z}_3$.

These are three similar problems on field extensions. Can anybody help me please – how can I solve this type of problem? I am learning about field extensions on my own, so my ideas are not very clear. Please help.

Best Answer

Hint: Given a field $K$ and some $f\in K[x]$, then $f$ has multiple roots (when considered over $\overline{K}$, an algebraic closure of $K$) if and only if $\gcd(f,f')\neq 1$, where $f'$ is the formal derivative of $f$.

This is mentioned in the Wikipedia page on "separable polynomial".

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