Some field theoretical properties and an application concerning transcendental numbers

Abstract

For a proper subfield $K$ of $\QQ$ we show the existence of an algebraic number $\alpha$ such that no power ${\alpha }^n$, $n\ge 1$, lies in $K$. As an application it is shown that these numbers, multiplied by convenient Gaussian numbers, can be written in the form $P(T{)}^{Q(T)}$ for some transcendental numbers $T$ where $P$ and $Q$ are arbitrarily prescribed non-constant rational functions over $\QQ$.

7 pages