Let α be a root of x3+x+1∈Z2[x] and β a root of x3+x2+1∈Z2[x]. Then we know that Z2(α)≃Z2[x](x3+x+1)≃F8≃Z2[x](x3+x2+1)≃Z2(β).
I need to find an explicit isomorphism Z2(α)→Z2(β).
I was thinking of finding a basis for Z2(α) and Z2(β) over Z2.
I let {1,α,α2} and {1,β,β2} be these two bases. Now suppose I have a field morphism ϕ:Z2(α)→Z2(β) which maps 1 to 1. how can I show that the image of α, i.e. ϕ(α), completely determines this map?
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