My book defines a new variable x_1 in terms of another x_0. It basically defines it as x_1 = (x_0)^{1/2}. Now we are working in an algebraically closed field K, with x_0 and x_1 in K. But what confuses me, and the author does not even mention, is that (x_0)^{1/2} is not a well-defined statement, yes, the square root exists because it is an algebraically closed field, but it is not necessarily unique.

So my question is: Let K be an algebraically closed field, what is the defintion of x^{1/2} for x in K?