Let F be a field of odd characteristic. Let α,β in F be non-zero. β does not have a square root in F and α has a square root in F[x]/(x2−β). Prove exactly one of α,αβ has a square root in F.
I've tried to express the elements in F[x]/(x2−β) using a basis {1,γ}, where γ2−β=0, but did not make a progress. In particular, I don't know how to use the condition of F having odd characteristic. Any help will be appreciated.
Answer
Hint: write α=(f1+f2γ)2 where γ2=β and fi∈F. See what this tells you about f1,f2.
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