In invariant theory the Reynold's Operator gives rise to an element invariant for that group.

For a Galois extension $K/F$ with $K=F[\alpha]$ the trace of $\alpha$ is an element of $K$. If $\alpha$ is of degree $n$, trace is of degree 1.

If I take sum of conjugates of $\alpha$ over a subgroup of index $m$ in the Galois group, and their product there seems to be asymmetry. The elements obtained are in different fields (that is their degrees are unequal.

Example: Take the $n$th cyclotomic extention over the rationals. Then the sum $\zeta+\bar \zeta$ gives rise to an element of degree $\phi(n)/2$. But their product $\zeta\bar\zeta=1$ is a rational number.

The reason $\zeta\mapsto\zeta^{-1}$ is an automorphism does not really explain what is happening.

I have three questions (at least for $F$ the field of rational numbers).

(1) Does this happen only in Galois extensions having complex conjugation as non-trivial automorphism?

(2) Are there examples where product over the (relative) conjugates giving rise to a higher degree element than the sum?

(3) Is there a classification of base fields $F$ where there the behaviour is symmetric for sum and products?