Question

1. Let K is a field, K[x] – the ring of all polynomials of one variable x over K (i.e. with coefficients in the field K) and f(x) is from K[x]. Prove that K[x]/(f) is a field if and only if the polynomial f is irreducible over K. For a given prime number p, construct a field with p² elements - the Galois filed GF(p²). Give an example.
2. Prove that for any prime p, (p - 1)! + 1 is divisible by p. (Wilson’s Theorem)
3. Prove that each element of the Galois field GF(2^n), n = 1, 2,… with even elements is a perfect square.
4. Let a prime p == 2 (mod 3). Prove that each element of the Galois field GF(p) is a perfect cube.

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Some Properties of the Finite Fields (Galois Fields)

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