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finite_field_primer

# finite_field_primer - A Primer on Finite Fields In(1(7 F...

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A Primer on Finite Fields In (1) - (7) F will denote a finite field. (1) F contains a copy of Z p = F p , for some prime p . (This prime is called the characteristic of F .) (2) There is a positive integer d with | F | = p d . Proof. From the definitions, F is a vector space over F p . Let e 1 , . . . , e d be a basis. Then F = n d i =1 a i e i a 1 , . . . , a d F p o . Thus | F | is the number of choices for the a i , namely p d . (3) Let α F F p . Then F p [ α ] = ( k X i =0 a i α i k 0 , a i F p ) is a subring of F . (That is, it is closed under addition, subtraction, and multiplication. See (6) and (8) , where we show that F p [ α ] is actually a subfield.) (4) Let m ( x ) F p [ x ] be a monic polynomial of minimal degree with m ( α ) = 0. (It exists since F is finite.) Then F p [ α ] is a copy of F p [ x ] (mod m ( x )), that is, the polynomial ring F p [ x ] with arithmetic done modulo the polynomial m ( x ). The polynomial m ( x ) is called the minimal polynomial of α over F p and is uniquely determined. We sometimes write m α ( x ) or even m α, F p ( x ) for the minimal polynomial of α over F p . Let f ( x ) be a nonconstant polynomial of K [ x ]. Then f ( x ) is called irreducible in K [ x ] if every factorization f ( x ) = a ( x ) b ( x ) in K [ x ] has { deg a, deg b } = { 0 , deg f } . (This corresponds to prime numbers in Z .) Otherwise f ( x ) is reducible .

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finite_field_primer - A Primer on Finite Fields In(1(7 F...

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