I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As the title suggests, it has to do with factorizing an otherwise irreducible expression into "polynomials" of algebraic degree.
Consider the binomial $x+y$. If I'm reading the definition correctly, then this is certainly an irreducible polynomial since we can't decompose it into polynomial pieces. We can $(\spadesuit)$, however, factorize it to obtain $$x+y=\left(x^{1/2}\right)^2-\left(i y^{1/2}\right)^2=\left(x^{1/2}+iy^{1/2}\right)\left(x^{1/2}-iy^{1/2}\right)$$
$(1)$ Under what circumstances is $(\spadesuit)$ a "legal move"? Do I need to be specific about what $\left(a^{1/n}\right)^n$ might mean?
$(2)$ Is there a name for this kind of "reducibility"?
We can continue in this manner to obtain the next iteration, $$x+y=\left(x^{1/4}\color{red}\pm i^{3/2}y^{1/4}\right)\left(x^{1/4}\color{red}\pm i^{1/2}y^{1/4}\right)$$ followed by $$x+y=\left(x^{1/8}\color{red}\pm i^{7/4}y^{1/8}\right)\left(x^{1/8}\color{red}\pm i^{5/4}y^{1/8}\right)\left(x^{1/8}\color{red}\pm i^{3/4}y^{1/8}\right)\left(x^{1/8}\color{red}\pm i^{1/4}y^{1/8}\right)$$ and so on, where I use $\color{red}\pm$ to mean $(a\color{red}\pm b)=(a+b)(a-b)$ just to save space.
$(3)$ Same as $(1)$ but for successive iterations.
Barring any errors thus far, we can arrive at a compact form relying on product notation: $$\large \prod_{\substack{n\in\mathbb{N}\\[.5ex]1\le r\le 2^{n-1}}}\left(x^{2^{-n}}\color{red}\pm i^{(2r-1)2^{-(n-1)}}y^{2^{-n}}\right)$$
$(4)$ Is there anything about this kind of manipulation that is blatantly wrong or otherwise doesn't hold up?
$(5)$ What fields of math, if any, would be interested in studying this kind of factorization?
In addition to being unsure how to frame this question properly, I'm also confuddled as to how to tag this question. Any suggestions/edits would be appreciated.