It is well known that if K is a number field whose discriminant is square free then K is monogenic.
I want to know if the converse is true. If K is monogenic then is the discriminant necessarily square free. In particular the contrapositive, if D is not square free then K is not monogenic.
Is this true?
The following converse is not true: "if $K$ is monogenic then its discriminant is squarefree". As an example, consider the cubic number field $$ K=\Bbb Q(\sqrt[3]{6}), $$ which is monogenic with basis $\{1,\alpha,\alpha^2\}$ for its ring of integers $\Bbb Z[\alpha]$, where $\alpha=\sqrt[3]{6}$. Its discriminant is given by \begin{align*} D(1,\alpha,\alpha^2) & =\det \begin{pmatrix} tr (1) & tr (\alpha) & tr (\alpha^2) \\ tr (\alpha) & tr (\alpha ^2) & tr (\alpha ^3) \\ tr (\alpha ^2) & tr (\alpha ^3) & tr(\alpha^4)\\ \end{pmatrix} \\[0.2cm] & = \det \begin{pmatrix} 3 & 0 & 0 \\ 0 & 0 & 18 \\ 0 & 18 & 0 \end{pmatrix} =-3\cdot 18^2=-972, \end{align*}
which is not squarefree. So the converse is false.