Editing Évariste Galois (section)

=== Continued fractions ===
In his first paper in 1828,<ref name="continuedfraction" /> Galois proved that the regular continued fraction which represents a quadratic [[Nth root|surd]] ''ζ'' is purely periodic if and only if ''ζ'' is a [[Periodic continued fraction#Reduced surds|reduced surd]], that is, <math>\zeta > 1</math> and its [[Conjugate (algebra)|conjugate]] <math>\eta</math>
satisfies <math>-1 < \eta < 0</math>.

In fact, Galois showed more than this. He also proved that if ''ζ'' is a reduced quadratic surd and ''η'' is its conjugate, then the continued fractions for ''ζ'' and for (−1/''η'') are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have

:<math>
\begin{align}
\zeta& = [\,\overline{a_0;a_1,a_2,\dots,a_{m-1}}\,]\\[3pt]
\frac{-1}{\eta}& = [\,\overline{a_{m-1};a_{m-2},a_{m-3},\dots,a_0}\,]\,
\end{align}
</math>

where ''ζ'' is any reduced quadratic surd, and ''η'' is its conjugate.

From these two theorems of Galois a result already known to Lagrange can be deduced. If ''r''&nbsp;>&nbsp;1 is a rational number that is not a perfect square, then

:<math>
\sqrt{r} = \left[\,a_0;\overline{a_1,a_2,\dots,a_2,a_1,2a_0}\,\right].
</math>

In particular, if ''n'' is any non-square positive integer, the regular continued fraction expansion of √''n'' contains a repeating block of length ''m'', in which the first ''m''&nbsp;−&nbsp;1 partial denominators form a [[palindrome|palindromic]] string.