現代数学の系譜工学物理雑談古典ガロア理論も読む78

現代数学の系譜工学物理雑談古典ガロア理論も読む78at MATH

現代数学の系譜工学物理雑談古典ガロア理論も読む78 - 暇つぶし2ch376: Polynomial factorization 6.1 Irreducible polynomials of a given degree 6.2 Number of monic irreducible polynomials of a given degree over a finite field 7 Applications 8 Extensions 8.1 Algebraic closure 8.1.1 Quasi-algebraic closure 8.2 Wedderburn's little theorem 8.3 Relationship to other commutative ring classes 9 See also Existence and uniqueness Let q = p^n be a prime power, and F be the splitting field of the polynomial Explicit construction Non-prime fields Given a prime power q = pn with p prime and n > 1, the field GF(q) may be explicitly constructed in the following way. One chooses first an irreducible polynomial P in GF(p)[X] of degree n (such an irreducible polynomial always exists). Then the quotient ring Relationship to other commutative ring classes Finite fields appear in the following chain of inclusions: commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields See also Quasi-finite field Field with one element Finite field arithmetic Finite ring Finite group Elementary abelian group Hamming space (引用終り)

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