Some field theoretical properties and an application concerning transcendental numbers

TL;DR

The paper explores properties of algebraic numbers relative to subfields of rationals and demonstrates how certain transcendental numbers can be expressed using prescribed rational functions, revealing new connections between algebraic and transcendental number theory.

Contribution

It establishes the existence of algebraic numbers outside any given subfield and shows how these can be represented as powers of transcendental numbers with prescribed rational functions.

Findings

Existence of algebraic numbers not contained in any proper subfield of Q.

Representation of these numbers as P(T)^{Q(T)} with transcendental T.

Application to expressing numbers as powers involving Gaussian and rational functions.

Abstract

For a proper subfield $K$ of $\QQ$ we show the existence of an algebraic number $\alpha$ such that no power $\alpha^n$, $n\geq 1$, 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 $\QQ$.