So far, I have the following:
Could someone show me why $h \mid x^q -x$ and also why $h$ has a root $b$ in $F$? I can figure out the rest. Thank you for your help!!
2026-04-08 00:40:49.1775608849
problem involving polynomial ring over a field
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Let $F_q$ be shorthand for the quotient.
Since $F_q\setminus\{0\}$ is a finite multiplicative group of order $q-1$, $a^{q-1}=1$ for all $a\in F_q\setminus \{0\}$. Then also, $a^q=a$ for these elements, and $0^q=0$ already.
So look: whatever element $\bar{x}$ happens to be in the field, $\bar{x}^q-\bar{x}=\bar{0}$, meaning that $\overline{x^q-q}=\bar{0}$, hence $x^q-x\in (h)$. Can you take it from here?