I'd say simply the k for which k=x^2. Doesn't have to be unique.
Are you sure there's no discussion involved about algebraically extending K?
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?
I asked my professor this some time ago he said the same thing. I just think the author (Washington) should have been more explicit in saying that it does not matter.
If you mean to form the simple extensions and where are the square roots (since there are in fact the isomorphic). But there is no point, because an algebraically closed field got no proper algebraic extension.Are you sure there's no discussion involved about algebraically extending K?