For almost all finite fields, there is a 3-primitive 1-normal polynomial of degree m with any prescribed last two coefficients, with a finite list of possible exceptions.
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$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed
For almost all finite fields, there is a 3-primitive 1-normal polynomial of degree m with any prescribed last two coefficients, with a finite list of possible exceptions.