We introduce a square root map on Sturmian words and study its
properties. Given a Sturmian word of slope $alpha$, there exists
exactly six minimal squares in its language (a minimal square does not
have a square as a proper prefix). A Sturmian word $s$ of slope $alpha$
can be written as a product of these six minimal squares: $s = X_1^2
X_2^2 X_3^2 cdots$. The square root of $s$ is defined to be the word
$sqrt{s} = X_1 X_2 X_3 cdots$. The main result of this paper is that
$sqrt{s}$ is also a Sturmian word of slope $alpha$. Further,
we characterize the Sturmian fixed points of the square root map, and we
describe how to find the intercept of $sqrt{s}$ and an occurrence of
any prefix of $sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2 X_2^2 cdots
X_n^2 = (X_1 X_2 cdots X_n)^2$ in the language of Sturmian words of
slope $alpha$ where the words $X_i^2$ are minimal squares of slope
$alpha$.

We also study the square root map in a more general
setting. We explicitly construct an infinite set of non-Sturmian fixed
points of the square root map. We show that the subshifts $Omega$
generated by these words have a curious property: for all $w in Omega$
either $sqrt{w} in Omega$ or $sqrt{w}$ is periodic. In
particular, the square root map can map an aperiodic word to a periodic
word.