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PolyAlg

# PolyAlg - A-138 A.2 A.2.1 Polynomial Algebra over Fields...

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A-138 A.2 Polynomial Algebra over Fields A.2.1 Polynomial rings over fields We have introduced fields in order to give arithmetic structure to our alphabet F . Our next wish is then to give arithmetic structure to words formed from our alphabet. Additive structure has been provided by considering words as members of the vector space F n of n -tuples from F for appropriate n . Scalar multiplication in F n does not however provide a comprehensive multiplication for words and vectors. In order to construct a workable definition of multipli- cation for words, we introduce polynomials over F . Let F be a field, and let x be a symbol not one of those of F , an indetermi- nate . To any n -tuple indeterminate ( a 0 , a 1 , a 2 , . . . , a n - 1 ) of members of F we associate the polynomial in x : polynomial a 0 x 0 + a 1 x 1 + a 2 x 2 + · · · + a n - 1 x n - 1 . In keeping with common notation we usually write a 0 for a 0 x 0 and a 1 x for a 1 x 1 . Also we write 0 · x i = 0 and 1 · x i = x i . We sometimes use summation notation for polynomials: d X i =0 a i x i = a 0 x 0 + a 1 x 1 + a 2 x 2 + · · · + a d x d . We next define F [ x ] as the set of all polynomials in x over F : F [ x ] F [ x ] = { X i =0 a i x i | a i F, a i = 0 for all but a finite number of i } . Polynomials are added in the usual manner: X i =0 a i x i + X i =0 b i x i = X i =0 ( a i + b i ) x i . This addition is compatible with vector addition of n -tuples in the sense that if the vector a of F n is associated with the polynomial a ( x ) and the vector b is associated with the polynomial b ( x ), then the vector a + b is associated with the polynomial a ( x ) + b ( x ). Polynomial multiplication is also familiar: X i =0 a i x i · X j =0 b j x j = X k =0 c k x k , where the coefficient c k is given by convolution: c k = i + j = k a i b j . This multi- plication is the inevitable consequence of the distributive law provided we require

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A.2. POLYNOMIAL ALGEBRA OVER FIELDS A-139 that ax i · bx j = ( ab ) x i + j always. (As usual we shall omit the · in multiplication when convenient.) The set F [ x ] equipped with the operations + and · is the polynomial ring in polynomial ring x over the field F . F is the field of coefficients of F [ x ]. coefficients Polynomial rings over fields have many of the properties enjoyed by fields. F [ x ] is closed and distributive nearly by definition. Commutativity and additive associativity for F [ x ] are easy consequences of the same properties for F , and multiplicative associativity is only slightly harder to check. The constant poly- nomials 0 x 0 = 0 and 1 x 0 = 1 serve respectively as additive and multiplicative identities. The polynomial ax 0 = a , for a F , is usually referred to as a con- stant polynomial or a scalar polynomial . Indeed if we define scalar multiplication constant polynomial scalar polynomial by α F as multiplication by the scalar polynomial α (= αx 0 ), then F [ x ] with polynomial addition and this scalar multiplication is a vector space over F , a basis for which is the subset { 1 , x, x 2 , x 3 , . . . , x i , . . . } . (Note that ( - 1) · a ( x ) is the additive inverse of the polynomial a ( x ).)
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PolyAlg - A-138 A.2 A.2.1 Polynomial Algebra over Fields...

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