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An approach to normal polynomials through symmetrization and symmetric reduction
An irreducible polynomial of degree is {\em normal}
over if and only if its roots satisfy the
condition , where
is the circulant determinant. By
finding a suitable {\em symmetrization} of (A multiple of
which is symmetric in ), we obtain a condition on the
coefficients of that is sufficient for to be normal. This approach
works well for but encounters computational difficulties when . In the present paper, we consider irreducible polynomials of the form
. For and , by an indirect method, we
are able to find simple conditions on that are sufficient for to be
normal. In a more general context, we also explore the normal polynomials of a
finite Galois extension through the irreducible characters of the Galois group.Comment: 28 page