Suppose a irreducible monic polynomial in $ \mathbb {F} _q $ and suppose that f has $ deg (f) $ nonzero roots in an extension E of $ \mathbb{F}_q $ The book wrote: "each root b of has its polynomial over F_q the polynomial $$f(x)^\frac{s}{deg(f)} = x^s-c_{1}x^s-1 + \ldots + (-1)^{s}c_s $$
How is the characteristic polynomial of a root defined b? the book also says that $c_1 = Tr_\frac{E}{\mathbb{F}_q}(b)$
I do not know what characteristic polynomial is associated with a root of another polynomial and much less how it found $ c_1 $ The book is "Finite Fields", content of Gaussian sums