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Finding a Primitive Element of GF(49)

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Find an irreducible monic polynomial of degree 2 over the field Z_7 whose root is a primitive element GF(49).

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Solution Summary

We show how to find a primitive element of the finite field GF(49).

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First we need a way to characterize all monic irreducible polynomials of degree 2 over Z_7. We claim that f(x) = x^2 + ax + b is irreducible if and only if D = a^2 - 4b is a quadratic nonresidue (QNR) mod 7, i.e. if and only if D is congruent to 3, 5, or 6 mod 7. This is true since the roots of f are (-a +/- sqrt(D)) / 2, which lie in Z_7 (and hence f is reducible over Z_7) if and only if D is a quadratic residue (QR) modulo 7.

Next we try monic quadratic irreducible polynomials with a = 0. These all have the form x^2 + b, where -b is a QNR mod 7, which is ...

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