4. Suppose K is a finite field, and let : (Z.+.0) → (K. +.0k) be the unique group homomorphism with (1) = 1k. a) Show that is a ring homomorphism. b) Show that ker=pZ for some prime p. p is called the characteristic of the field K. c) Show that induces an injective ring homomorphism : F→ K. d) Deduce that K is a vector space over Fp. e) Conclude that K-p" for some n.

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