Dispersive and explosive mappings

Extract

Let Q, R be rational numbers and real numbers respectively. We use V(F) and W(F) to denote finite dimensional inner product spaces over F. Given V(Q), we use V(R) for the smallest inner space over R containing V(Q). It is known that an R-homomorphism of V(R) to W(R) is continous. We prove that if a Q-homomorphism f: V(R) → W(R), then f is dispersive, i.e., given any v₀ ∈ V(Q) and ε > 0, the image set f[D(v₀, ε)], where D(v₀, ε) = [v: v ∈ V(Q), ¦v – v₀¦ < ε], is not bounded. It is also shown that some Q-homomorphism f: V(Q) → W(Q) can be explosive in the sense that for any v₀ ∈ V(Q) and ε > 0, the set f[D[v₀, ε)] is dense in W(Q). As a particular case of dispersive and explosive Q-homomorphisms, we show that the algebraic number field isomorphism f: Q(a) → Q(β), where f(a) = β and α ≠ β or βmacr; (βmacr; being complex conjugates of β) is explosive.