Explicitly Constructing Finite Fields
Given a prime power q = pn, we may explicitly construct a finite field with q elements as follows. Select a monic irreducible polynomial f(T) of degree n in Fp. (Such a polynomial is guaranteed to exist, once we know that a finite field of size q exists: just take the minimal polynomial of any primitive element for that field over the subfield Fp.) Then Fp/(f(T)) is a field of size q. Here, Fp denotes the ring of all polynomials in T with coefficients in Fp, (f(T)) denotes the ideal generated by f(T), and the quotient is meant in the sense of quotient rings — the set of polynomials in T with coefficients in Fp modulo (f(T)).
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