rational roots of a cubic

Question

If $a$ and $b$ are rational numbers, is there any if and only if condition on them, so that $x^3+ax+b=0$ will have no rational roots? The answer gives a condition that splitting field of the same polynomial over Q is of degree exactly 3. Thanks!!

Answer 1

You can apply the rational root test. And the conclusion is not necessarily trule. For example $x^3+2$ has no rational roots but the degree of the splitting field has degree 6.