Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree of its minimal polynomial. There are known algorithms that allow to easily compute sum and product of algebraic numbers in this representation, raise them to a rational power, extract real and imaginary parts, compare them, or evaluate them numerically to an arbitrary precision.
Is there an efficient algorithm that given an algebraic number $\alpha$ in this representation can decide if $\alpha$ can be represented as a sum or product of algebraic numbers of smaller degree?
 A: Just a few comments to get the discussion started.


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*I don't think that this problem for sums is very meaningful. Every element in ${\mathbb Q}(\sqrt{a},\sqrt{b})$ is a sum of three elements in proper subfields,
and an element in ${\mathbb Q}(\sqrt[4]{m})$ is a sum of elements in proper subfields if and only if it is already an element of  ${\mathbb Q}(\sqrt{m})$,
with the exception of a few cases where the extension happens to be bicyclic.

*The problem is more interesting for products. Here the first idea would be writing the ideal $(\alpha)$ of the given element $\alpha$ as a product of prime ideals and then checking whether this product may be written as a product of ideals living in subfields. If $(\alpha)$ is a product of ideals living in subfields then several principal ideal test will show whether they are a product of elements from subfields, up to a unit upstairs. This reduces the problem to writing a unit as a product of units coming from subfields.
My guess is that the latter problem can indeed be solved efficiently by looking at all the real and complex embeddings of the units in the subfields. 
