Editing Rolle's theorem (section)

== 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 [[field (mathematics)|fields]], but the following corollary does: if a real polynomial factors (has all of its roots) over the real numbers, then its derivative does as well. One may call this property of a field '''Rolle's property'''.{{citation needed|date=September 2018}} More general fields do not always have differentiable functions, but they do always have 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. Any algebraically closed field such as the [[complex numbers]] has Rolle's property. However, the rational numbers do not{{snd}} for example, {{math|1=''x''<sup>3</sup> − ''x'' = ''x''(''x'' − 1)(''x'' + 1)}} factors over the [[rational numbers|rationals]], but its derivative,
<math display="block">3x^2-1 = 3 \left(x - \tfrac{1}{\sqrt 3} \right) \left(x + \tfrac{1}{\sqrt 3} \right),</math>
does not. The question of which fields satisfy Rolle's property was raised in {{Harvnb|Kaplansky|1972}}.<ref>{{ Citation | first = Irving | last = Kaplansky | author-link = Irving Kaplansky | title = Fields and Rings | year = 1972 }}.{{full citation needed|date=July 2024}}</ref> For [[finite field]]s, the answer is that only {{math|'''F'''<sup>2</sup>}} and {{math|'''F'''<sup>4</sup>}} have Rolle's property.<ref>{{ Citation | title = Multiplier sequences for fields | first1 = Thomas | last1 = Craven | first2 = George | last2 = Csordas | journal = Illinois J. Math. | volume = 21 | year = 1977 | pages = 801–817 | url = http://projecteuclid.org/euclid.ijm/1256048929 | issue = 4 | doi = 10.1215/ijm/1256048929 | doi-access = free | url-access = subscription }}.</ref><ref>{{ Citation | title = A Simple Proof of Rolle's Theorem for Finite Fields | first1 = C. | last1 = Ballantine | first2 = J. | last2 = Roberts | journal = [[The American Mathematical Monthly]] | volume = 109 |date=January 2002 | pages = 72–74 | issue = 1 | doi = 10.2307/2695770 | jstor = 2695770 | publisher = Mathematical Association of America }}.</ref>

For a complex version, see [[Voorhoeve index]].