Baby Rudin Theorem 1.21 says the following:
For every real $x>0$ and every integer $n>0$ there exists a unique positive real $y$ such that $y^n = x$.
Essentially, this theorem tells us that every positive (trivially we could modify this to say nonnegative since the 0 case is easy) real has a unique positive real as an $n$th root. My question is, how do we modify or generalize this? From familiarity with the reals before making things rigorous (as in Baby Rudin), we can say for example that negative numbers have a unique $n$th root if $n$ odd. I'm sure there are other statements about roots that might be of interest and which I am missing here.
Thus my question is, what is the general theorem which captures all of these theorems under one rubric as special cases? Presumably this general theorem makes claims about existence and uniqueness as appropriate. It also seems likely to me that the general statement is actually in $\mathbb{C}$? At any rate, that the statement would make claims about existence and uniqueness are (I think) the key criteria which I am looking for in said theorem.
Basically, we can describe how (single-variable) polynomials factor over $\mathbb{R}$ completely via the following two facts:
Every polynomial factors into at-worst-quadratic factors.
For every real number $r$, exactly one of $r$ and $-r$ has a square root.
This is the "reals-only" equivalent of the fundamental theorem of algebra. Part of the compelling beauty of $\mathbb{C}$, though, is the clarity it provides (via the FTA as usually stated) in contrast to the above principle which in contrast seems a bit ad hoc. Yes, the latter is sufficient if you just want to look at the real numbers, but in some sense it's not the "right" story.
More broadly, I think you're looking for the notion of a real closed field.