Let $K$ be a field and $K(u)$ be an algebraic extension of $K$. Now $u$ satisfies an irreducible polynomial over $K$. Does that mean $u$ can be written as algebraic combinations of elements of $K$?
In the first place what does expressing something as algebraic combinations mean ? This question might be a trivial one but I am having some conceptual doubt regarding the meaning of algebraic combinations ! I would appreciate if the definition is elaborated and some remarks are made.
Thank You !
In general, an "algebraic combination" of some objects is just a polynomial expression in these objects. So if $K$ is a field, an algebraic combination of elements of $K$ is also an element of $K$.
So no, $u$ cannot be written as such a combination unless $u\in K$.