Use a purely group-theoretic argument to show that if F is a field of order p^n, then every element of F* is a zero of x^p^n-x
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F* is a group of order p^n - 1 and identity element 1.
Therefore for any element x in F*,
x^(p^n - 1) = 1
Multiplying both sides by x, and then subtracting x from both sides,
x^(p^n) = x.
x^(p^n) - x = 0
In other words, every element of F* is a zero of the polynomial
x^(p^n) - x.
Therefore for any element x in F*,
x^(p^n - 1) = 1
Multiplying both sides by x, and then subtracting x from both sides,
x^(p^n) = x.
x^(p^n) - x = 0
In other words, every element of F* is a zero of the polynomial
x^(p^n) - x.