Generalizations To Other Fields
Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field. As such, it does not generalize to other fields, but the following corollary does: if a real polynomial splits (has all its roots) over the real numbers, then its derivative does as well – one may call this property of a field Rolle's property. More general fields do not always have a notion of differentiable function, but they do have a notion of polynomials, which can be symbolically differentiated. Similarly, more general fields may not have an order, but one has a notion of a root of a polynomial lying in a field.
Thus Rolle's theorem shows that the real numbers have Rolle's property, and any algebraically closed field such as the complex numbers has Rolle's property, but conversely the rational numbers do not – for example, splits over the rationals, but its derivative does not. The question of which fields satisfy Rolle's property was raised in (Kaplansky 1972). For finite fields, the answer is that only and have Rolle's property; this was first proven via technical means in (Craven & Csordas 1977), and a simple proof is given in (Ballantine & Roberts 2002).
For a complex version, see Voorhoeve index.
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