Question

Thee part question. Please answer all parts!

Let E be a field of characteristic p > 0 (we proved p must always be prime). Verify that the ring homomorphism X : Z → E determined by sending χ : 1-1 E (the unity in E) ( so x(n)-n 1E wheren1E 1E 1E (n-times), x(-n)- nle for any n 1,2,3,... and X(0) 0E by definition of χ) is in fact a ring homomorphism with ker(X) = pZ. Úse the fundamental homomorphism theorem for rings to conclude that Zp-pz Imx. Show that if F is a subfield of a field E then the abelian group (E, +) is a vector space over F where scalar multiplication is defined, for any λ E F and any γ E E, by λ . γ = λγ, their product in E. Using the above construction, let E be a finite field of characteristic p. By question 69 we have that E contains (an ismorphic copy of) Zp. By question 70 we have E is a vector space over Zp. Let γ1, , γη be a basis for E, as a Zp ve(tor space and define a linear map T : E → linear isomorphism between E and (Zp)". Le. show that T is bijective and linear.

Answer 1

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