Polynomial spaces over finite fields Original Research Article

Burdeʹs theory about p-dimensional vectors modulo p (J. Reine Angew. Math. 268/269 (1974) 302–374, 278/279 (1975) 353–364) is generalized to a theory of q-dimensional vectors over an arbitrary finite field GF(q). We interpret the q-dimensional space over GF(q) as the space of polynomials with degree less than q. The connection between the monomials imagepk(x)=xkset membership, variantGF(q)[x]; k=0,…,q−1,and the linear mapsimagephia(f(x))=f(x+a); f(x)set membership, variantGF(q)[x], aset membership, variantGF(q),governs the transition from the additive to the multiplicative structure of GF(q). The investigation of the subspaces Uphi(pk)={phia(pk)aset membership, variantGF(q)} gives some relations between the pk which can be interpreted as results on complex characters.