= Squarefree reduction excludes a prime from the power-basis index
{title2=$\overline f\text{ squarefree}\implies p\nmid[\mathcal O_K:\mathbb Z[\alpha]]$}
For an <algebraic integer> $\alpha$ generating a <number field>, the <polynomial discriminant> of its <minimal polynomial> $f$ equals the <discriminant of elements of a number field> for its power basis. A squarefree reduction modulo $p$ has nonzero <resultant> with its derivative, so $p\nmid\operatorname{disc}(f)$. The <discriminant-index formula for an integral lattice>, $\operatorname{disc}(f)=[\mathcal O_K:\mathbb Z[\alpha]]^2d_K$, proves the assertion.
// Target: arithmetic.bigb
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