In this diploma thesis we first look at Hilbert Nullstellensatz, along with some examples. The main focus of this work, however, is combinatorial nullstellensatz by Alon, which requires stricter conditions and provides a stronger result. We also look at its corollary, which turns out to be a powerful tool when proving some already known theorems. We present detailed proof of both the Combinatorial Nullstellensatz and its corollary in this work. Afterwards, we look at some cases where we can use the combinatorial nullstellensatz to prove already known theorems from different fields of mathematics, such as the Chevalley-Warning theorem of common zeros of a family of polynomials, the Cauchy-Davenport theorem of cardinality of two nonempty subsets of Zp, and some other examples from geometry and graph theory.
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