TITLE: What is the meaning of "algebraically indistinguishable"
QUESTION [6 upvotes]: I heard the term couple of times (in Field theory class and book),
for example: The different roots of $p(x)=x^3-2$ are "algebraically indistinguishable".
I understand the meaning intuitively, but what does it really mean ?

REPLY [2 votes]: Which of the two square roots of $−1$ should we call $i$ and which should we call $−i$? Whichever one we call $i$ we will show one unit above $0$ when we draw a picture. Does it make any sense to say there's a difference between choosing one of them for that role rather than the other? If not, then in that sense they're indistinguishable.