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A minimal clone is an atom of the lattice of clones. A minimal function is, briefly saying, a function which generates a minimal clone. For a prime power k we consider the base set with k elements as a finite field GF(k). We present binary idempotent minimal polynomials and ternary majority minimal polynomials over GF(3) and generalize them to minimal polynomials over GF(k) for any prime power k ges3.