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Trace formulae for irreducible polynomials over $$\mathbb F_P$$ with minimal order roots in $$\mathbb F_{P^q}$$. (English) Zbl 1178.11074
Let $$P$$ be a prime of the form $$P = q^ns+1$$ for a prime $$q$$. Then $$P^q = q^{n+1}sK+1$$, where $$\gcd(K,P-1) = 1$$. The author gives formulas involving values of the trace function $$\text{Tr}: \mathbb F_{P^q}\to\mathbb F_P$$ of elements $$\alpha \in \mathbb F_{P^q}$$ of order $$R$$ for a prime divisor $$R$$ of $$K$$. For instance $$\text{Tr}(\alpha)+\text{Tr}(\alpha^{-1}) = -1$$, $$\text{Tr}(\alpha)\text{Tr}(\alpha^{-1}) = (q+1)/2$$ if $$q > 2, R = 2q+1$$ (take e.g. $$P=401$$, $$q=5$$, $$R=11$$).
##### MSC:
 11T06 Polynomials over finite fields
##### Keywords:
Trace-function; minimal polynomial; reciprocal polynomial
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##### References:
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