Six-element vector space
Task number: 2508
Is it possible for the structure \((\{0{,}1,2{,}3,4{,}5\},\oplus,\odot)\) to form a vector space over the field \(\mathbb Z_3\), where \(\mathbf u \oplus \mathbf v = \mathbf u+\mathbf v \mod 6\) and \(a\odot \mathbf u= a\cdot \mathbf u \mod 6\)?
Resolution
It is impossible, because for example \(2\odot 3=0\) which is in conflict with axioms of a vector space.
