Let $a, b$ be two algebraic numbers, $f(x), g(x)$ their minimal polynomials over $\mathbb{Q}$ respectively. Suppose the set of roots of $f$ is $$A = \{a=a_1, a_2, \cdots, a_n\}$$ and the set of roots of $g$ is $$B = \{b=b_1, b_2, \cdots, b_m\}.$$
According to this wikipedia page - other applications, there is a way to build a polynomial, one of whose roots being $a+b$, from the coefficients of $f$ and $g$. Explicitly, the polynomial in $x$ is $res_y(f(y),g(x-y))$.
However, it is by no means the minimal polynomial of $a+b$. This is not surprising, since the coefficients of $f$ and $g$ only encode the unordered sets A and B, in which $a$ and $b$ are not specified in any way.
Hence, instead of looking at the element $a+b$, I look at the set $S = \{a_i+b_j|a_i\in A, b_j\in B\}$ and come up with two questions.
Questions
How do the Galois groups of $a$ and $b$ relate to the Galois groups of the elements in $S$?
Which elements in $S$ have the same minimal polynomials?
Is the polynomial (in $x$) $res_y(f(y),g(x-y))$ the product $P$ of the minimal polynomials in the last question? If not, is there a general description to get $P$ from the coefficients of $f$ and $g$?
Examples
Let $$a=\sqrt{2}, b=\sqrt{3}.$$ Then $$f=x^2-2, g=x^2-3$$ and $$A=\{\sqrt{2}, -\sqrt{2}\}, B=\{\sqrt{3}, -\sqrt{3}\}.$$ The polynomial obtained by the resultant is $$res_y(f(y),g(x-y)) = x^4 - 10x^2+1,$$ which is irreducible whose set of roots is $$S = \{\pm\sqrt{2}\pm\sqrt{3}\}.$$
In this case the Galois group of $a+b$ is isomorphic to the product of the Galois groups of $a$ and $b$ (1). All four elements in $S$ share the same minimal polynomial (2). And the minimal polynomial of $a+b$ is obtainable from the coefficients of $f$ and $g$ by the resultant formula (3).