In many settings,
- the square function $x \mapsto x^2$ is mostly two-to-one, except at zero
- given $y$ in some ring $R$ with a product operation, there can be zero,
one or two values of $x$ for which $x^2 = y$ (i.e. $x \cdot x = y$).
When two such $x$ exist, they are opposites of each other,
and called "the square roots of $y$"
- in many cases there is a way to extend the ring $R$ to a bigger ring $S$
where all nonzero elements have two square roots. For example if $R$ is
the field $\mathbb{R}$ of real numbers then we can take $S$ to be the
field $\mathbb{C}$ of complex numbers
When working with the real numbers,
- zero is its own unique square root
- positive numbers have two square roots in the real numbers
- negative numbers have no square root in the real numbers,
but two square roots in the complex numbers
The two square roots of a positive number being opposite to each other,
one is negative and one is positive. Since we find positive numbers more
"natural", we often decide to give the positive root a special role and
to call it "square root of $y$", which could be thought of as short for
"the positive square root of $y$". The other root is then referred to
as "minus square root of $y$".
In computer algebra systems, one has to decide what to do when the
user asks for square roots a command such as sqrt(y) or y.sqrt():
- regarding where to look for square roots:
- stay in the ring where $y$ lives
- or extend it, if necessary, to a larger ring where square roots exist,
if they did not in the ring where $y$ lives
- regarding how many square roots to return:
- return all the square roots of its argument,
- or return one particular square root
(e.g. the positive one when that makes sense)?
I suppose the documentation you are referring to is the
SageMath documentation for sage.structure.element.CommutativeRingElement.sqrt.
It explains that y.sqrt() takes two optional parameters,
extend to decide whether to look for square roots in a
larger ring if necessary, and all to decide whether to
return all square roots or a preferred square root when
there is one. There's an additional optional parameter
name to name the extra generator of the new ring if you
decide to extend.
You should provide the construction of
J,Candfso that people can reproduce your situation.