On Certain q-Polynomials

Zusammenfassung

Let p be a prime number, q a power of p, F _q the finite field of q elements and R any field containing F _q . A polynomial L(X) ∈ R[X] of the form

$$ L(X) = {c_n}{X^{{q^n}}} + {c_{n - 1}}{X^{{q^{n - 1}}}} + ... + {c_1}{X^q} + {c_0}X $$

(1)

is called a q-polynomial. Note that every q-polynomial is also a p-polynomial. We assume throughout that L is monic (c _n = 1) and c₀≠ 0. Since L’(X) = c₀≠ 0, this guarantees that L(X) has q ⁿ distinct roots (in any splitting field). If X,Y are indeterminates and a,b ∈ F _q then L(aX + bY) = aL(X) + bL(Y); thus the mapping x → L(x) of R into itself is a linear transformation of R, considered as a vector space over F _q . For this reason such polynomials are also called ‘linearized’ polynomials. If S is any extension field of R in which L(X) splits, the F _q -linearity of L as a transformation of S shows that the set U of roots of L in S is an F _q -space (for U in the kernel of L). Since ∣U∣ = q ⁿ , U is an n-dimensional F _q -space. Conversely, starting with a given n-dimensional F _q -subspace U of R it can be shown that the polynomial L(X)=Π_u∊U (U-u) is indeed a q-polynomial of the shape (1); we write it as L(X; U) to indicate the dependence on U. For a proof of this fact and further information about q-polynomials see [3]. Although that book is concerned only with finite fields note that Lemma 3.51 and Theorem 3.52 on pp. 109–110 hold when F _qm is replaced by any field R containing F _q .