I have been attempting to read this paper by Alexander Aptekarev. In it, he proves that $\delta$ and $\gamma$ cannot both be rational simultaneously. He also notes that this result follows from a paper by A.B. Shidlovski, in which it was shown that $$1-\frac{1}{e}\ \ \text{and}\ \ -\left(\gamma-\frac{\delta}{e}\right)$$ are algebraically independent. This is the terminology I am hung upon. Although I am familiar with what an algebraic number is (and hence what transcendental numbers are), I can't seem to wrap my head around what it means for two numbers to be algebraically independent, or from such a statement we can deduce that $\delta$ and $\gamma$ cant both be rational.
When I try to look up algebraic independence, the definitions use terminology that goes way over my head (as I am simply a math undergrad). Is there a way of explaining this concept to a novice like myself in a way that I can understand how from such a statement I can deduce that $\delta$ and $\gamma$ cant both be rational? All insight is appreciated.