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As stated, what would be an example to satisfy the above criterion?, what would be an example of a field $k$ of characteristic $p$, and an irreducible, inseparable polynomial in $k[x]$

Tony2055
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    The standard example is $k=K[T^p]$ for a field $K$ of characteristic $p$ and the polynomial $X^p-T^p$. – Vercassivelaunos Apr 19 '21 at 05:32
  • A class of examples is given here: https://math.stackexchange.com/questions/403924/xp-c-has-no-root-in-a-field-f-if-and-only-if-xp-c-is-irreducible?noredirect=1&lq=1 – 19021605 Jun 24 '25 at 11:12

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Hint: For $p $ prime, let $K=\mathbb F_p (T) $, the field of rational polynomials in an indeterminate $T $. Then consider the polynomial $q (x)=x^p-T\in K[x] $.

It's not separable because $q'(x)=0$ ( so it can't be relatively prime to it's derivative). Or, we have for any root $\alpha$ that $q(x)=(x-\alpha)^p$ (freshman's binomial), so the root is repeated.

But it's irreducible. For that you can use Eisenstein.

  • Can you be a little bit more specific about an indeterminate $T$? – Tony2055 Apr 19 '21 at 06:10
  • Deep question. It's basically a variable, or an undefined entity outside the field $\mathbb F_p$. Like an unknown. –  Apr 19 '21 at 06:40
  • So is $x^5-10$ would be an example for this if we just look at it in $\Field_5$? – Tony2055 Apr 20 '21 at 03:48
  • No. Rather $x^5-T$. Remember we are working in $\mathbb F_5(T)[x]$. $10\equiv0$ in $\mathbb F_5$, so your example would just be $x^5$, which is not irreducible. –  Apr 20 '21 at 04:02