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I'll try to give additional explanation on algebraic number and the link with EC. Let $q=2^{163}$ the finite Field $F_q=GF(q)$, as selected by NIST has some features for doing cryptography. This field has been generated with the irreduccible pentanomial you gave in the table. To understand what the trace of the polynomial is, it corresponds to the ...


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Actually, you don't compute the trace of a polynomial per se, but of a finite-field element, which is expressed as a polynomial-like expresssion with $u$ acting as the indeterminate. ($u$, as you are probably aware, is a root of the generating polynomial $p(t)$.) Mathematically, the trace of $u$ is $$\mathrm{Tr}(u) = u + u^2 + u^4 + \cdots + u^{2^{162}}$$ ...



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