I am dealing with a computation that yields numbers that are sums of radicals of the following form:
$N=\sum_{i=0}^{m}{a_i\sqrt{b_i}}$
Where $a_i,b_i \in \mathbb{Q}$ (rationals). The context is intersection of propagating planes, and centers of inscribed spheres to tetrahedrons (as far as I can tell, they cannot be more simple than a sum of radicals).
I am interested whether each such number can be represented in the most simplest and, most important, unique form (s.t. two numbers $N_1,N_2$ are equal iff $N_1-N_2$ is uniquely represented by $0$). It is easy to show that in fact $b_i \in \mathbb{N}$ since the fraction can be pulled out into $a_i$. But how do I check if a sum is in its simplest form otherwise? (which is the same as asking how to simplify it further).
By simple, I mean that every radical is simple (no squares inside the square root), and that the sum has the fewest amount of summands.
My purpose is it to build a number type of such sums of radicals, for exact computations of the mentioned above. For this, I need it to be an algebraic field.
Side question 1: how to test if $N$ is positive or negative as quickly as possible?
Side question 2: is there an existing (coded) number type used in practice? I couldn't find any by extensive googling.
Side question 3: is the inverse to a radical sum always a radical sum (can be expressed by one)?
