Question

A finite field is any finite extension of F_p := Z/pZ. The characteristic

of a field F is the generator of the kernel of the map ι : Z → F, ι(1) = 1.

1. (a) Prove that there exist finite fields of order pⁿfor any prime p. We denote such

   a field F_pⁿ.

2. (b) Prove that F_pⁿ has characteristic p.

3. (c) Prove that the Frobenius map φ(a) = ap is an automorphism of F_pⁿ .

4. (d) If f(x) ∈ F_pⁿ [x], prove that f(x^p) is neither irreducible nor separable, i.e.,

   factors and has repeated roots.

Answer 1

a) 나- sch一千, rnt.df-E.。 ^ ld- is eb.ds ras ゜ー) is alio 七 or-t

pr Siner Simala pr hh sm. lai m 0) ヒ トー&a) 2イ 中

aloin フ) φ i's 1-1. ldl hanes hanes s -1 MY)2 ar pe rostチ子(nr)