I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
We consider polynomials f ∈ ℤ[x] with roots in ℝ, and for each polynomial f, the principal root is the real root with the largest magnitude. In the case of two roots of equal magnitude, we take the positive one.† So, for instance, √5 is the principal root of x2 − 5, the golden ratio φ is the principal root of x2 − x − 1, and −φ is the principal root of x2 + x − 1.
[ † Edit: Previously, I had defined the principal root to be the maximal one; I revised this definition based on remarks by Kevin Buzzard below. This also motivates some revisions to the questions I ask. ]
It's tempting to think that we could represent algebraic numbers by (minimal) polynomials over ℤ for which they are the principal root. We implicitly do this with rational numbers all the time: a/b denotes the real number which is the principal root of bx − a, and can formally be defined in such terms. This is also precisely what we do with algebraic integers: n√b is defined to be the principal root of xn − b, at least for b non-negative. This approach seems problematic for negative algebraic numbers such as φ−1, whose minimal polynomial is x2 + x − 1, which is the same as for −φ; thus every polynomial for which φ−1 is a root will also have the (larger in magnitude) root −φ. A similar problem arises for −√5, of course. But let us focus on algebraic numbers which are the principal roots of their minimal polynomials.
For two irredicible polynomials f,g ∈ ℤ[x] with roots in ℝ, let u,v ∈ ℝ be their principal roots.
- Are there broad classes of polynomials f and g, including ones of degree 2 or more, of course, for which there is a "simple" formula (e.g. involving no recursive functions more complicated than sums, products, exponents, and "well-known" number sequences) for the minimal polynomials of u + v, uv, or uv−1?
- Does there exist such a "simple" formula for some polynomial (not necessarily irredicuble) for which u + v, uv, or uv−1 is the principal root?
- If such questions are a proper subject of some body of research or well-studied theory: what is the name of the associated field of mathematics? (E.g. is this a special topic of Galois theory?)
Note that it is unlikely that we can obtain any sort of satisfactory answer for obtaining a minimal polynomial for u − v ; for instance, if u = √2 and v = √3, then the minimal polynomial of all four numbers ± √3 ± √2 have the same minimal polynomial, x4 − 26x2 + 145. For similar reasons, it is unlikely that there is a complete solution for arbitrary sums of principal roots: if u is a principal root of f(x), then −u is a principal root of f(−x), which differs from f in the case that f is not an even function of x. Thus, some differences of principal roots may also be expressed as sums of principal roots.
The difficulties described above and in the preamble suggest that a clean and elegant theory is unlikely; but I'm hoping that there are interesting classes of algebraic numbers which may be treated in this way.