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j.c.
j.c.
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To add to Keith's answer, there are various classes of number fields which are known to be not monogenic. For instance, the following paper

Marie-Nicole Gras, Non monogénéité de l'anneau des entiers des extensions cycliques de $\mathbb{Q}$ de degré premier $l\ge 5$Non monogénéité de l'anneau des entiers des extensions cycliques de $\mathbb{Q}$ de degré premier $l\ge 5$, J. Number Theory 23 (1986), 347-353

gives an elegant proof of the fact that no cyclic extension $K$ of the rationals of prime degree $l\ge 5$ is monogenic unless it happens to be the real part of a cyclotomic field.

To add to Keith's answer, there are various classes of number fields which are known to be not monogenic. For instance, the following paper

Marie-Nicole Gras, Non monogénéité de l'anneau des entiers des extensions cycliques de $\mathbb{Q}$ de degré premier $l\ge 5$, J. Number Theory 23 (1986), 347-353

gives an elegant proof of the fact that no cyclic extension $K$ of the rationals of prime degree $l\ge 5$ is monogenic unless it happens to be the real part of a cyclotomic field.

To add to Keith's answer, there are various classes of number fields which are known to be not monogenic. For instance, the following paper

Marie-Nicole Gras, Non monogénéité de l'anneau des entiers des extensions cycliques de $\mathbb{Q}$ de degré premier $l\ge 5$, J. Number Theory 23 (1986), 347-353

gives an elegant proof of the fact that no cyclic extension $K$ of the rationals of prime degree $l\ge 5$ is monogenic unless it happens to be the real part of a cyclotomic field.

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Robin Chapman
Robin Chapman
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To add to Keith's answer, there are various classes of number fields which are known to be not monogenic. For instance, the following paper

Marie-Nicole Gras, Non monogénéité de l'anneau des entiers des extensions cycliques de $\mathbb{Q}$ de degré premier $l\ge 5$, J. Number Theory 23 (1986), 347-353

gives an elegant proof of the fact that no cyclic extension $K$ of the rationals of prime degree $l\ge 5$ is monogenic unless it happens to be the real part of a cyclotomic field.