Saturday, 10 August 2019

abstract algebra - Square root in finite degree extension of odd characteristic field.



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]/(x2β). Prove exactly one of α,αβ has a square root in F.



I've tried to express the elements in F[x]/(x2β) using a basis {1,γ}, where γ2β=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 fiF. See what this tells you about f1,f2.



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