Finite fields
Combinatorics, Probability and Computing
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
On Polynomials over Prime Fields Taking Only Two Values on the Multiplicative Group
Finite Fields and Their Applications
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Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a"1,...,a"n of GF(q), there are distinct field elements b"1,...,b"n such that a"1b"1+...+a"nb"n=0. This implies the classification of hyperplanes lying in the union of the hyperplanes X"i=X"j in a vector space over GF(q), and also the classification of those multisets for which all reduced polynomials of this range are of reduced degree q-2. The proof is based on the polynomial method.