Is it true that when a root of a 4th degree polynomial is natural, the radicals inside the formula are always rational?

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Given the formula for the 4th degree polynomial, is it true that a root is a natural only when all the radicals inside the formula are rational numbers?

Edit1: The coeficients are whole numbers.

https://upload.wikimedia.org/wikipedia/commons/9/99/Quartic_Formula.svg