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## Secondary language

Language: English Finite fields The topic, presented in this Master's thesis, are structures in the field of algebra - finite fields. We first introduce basic definitions and ring and group features to facilitate the understanding of finite fields. After that we discuss the fields. A field is a commutative ring with the unit 1≠0, where every nonzero element is inversible. Every field is an integral domain, which's characteristic is either 0 or p, where p is a prime number. The K extension of the field F is finite, if a field K, that we consider as a vector space over the field F, is finite dimensional. If the finite field F has q elements and K is the finite extension of the field F, than K has q^n elements, where n =[K:F]. If K is the extension of the field F and f(x)∈F[x] is a nonconstant polynomial, that splits in the field K and doesn't split in none of the proper subfields of field K, than K is the splitting field of the polynomial f(x) over F. We prove, that any two fields with finite number of elements that are also splitting fields of the polynomial f(x)=x^(p^n)-x over ℤ_p, are isomorphic. Complying with those claims the characterization of the finite fields follows, which states, that for any prime number p and any positive integer n, there is, up to isomorphism, a unique finite field of p^n elements. In the end we discuss one of the basic theorems in this thesis, the Wedderburn's theorem, which states that every finite division ring is a field. Characteristic of a field, characteristic of a ring, classification of finite field, extension field, factor ring, field, finite field, ideal, ideal domain, maximal ideal and prime ideal, polynomial ring, ring homomorphism, splitting field, vector space, Wedderburn's theorem, zeros of polynomial.