Problem
ALG-B3-M10-P005 Square Root of Two on the Rationals
#5
★★★★☆ Level 4 of 5
Prove that there is no additive \(f:\mathbb Q\to\mathbb Q\) such that \(f(f(x))=2x\).
The coefficient would need square \(2\).
If \(f(x)=cx\), then \(f(f(x))=c^2x\). We need \(c^2=2\), but no rational \(c\) has this property. Contradiction.
The domain \(\mathbb Q\) creates impossibility.