By Antonio Machì (auth.)
This e-book offers with a number of issues in algebra worthy for laptop technological know-how purposes and the symbolic remedy of algebraic difficulties, mentioning and discussing their algorithmic nature. the themes coated diversity from classical effects corresponding to the Euclidean set of rules, the chinese language the rest theorem, and polynomial interpolation, to p-adic expansions of rational and algebraic numbers and rational features, to arrive the matter of the polynomial factorisation, specifically through Berlekamp’s procedure, and the discrete Fourier remodel. easy algebra suggestions are revised in a kind suited to implementation on a working laptop or computer algebra system.
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Additional info for Algebra for Symbolic Computation
A2k = Ak , for all k. (Formulas 2. and 3. ) The matrix A represents a linear transformation of a vector space V . , and 3. allow us to decompose the space into a direct sum of subspaces. , v = Iv = A0 v + A1 v + · · · + Am v, so V = A0 (V ) + A1 (V ) + · · · + Am (V ), and V is the sum of the subspaces Ai (V ). Moreover, this sum is direct. Indeed, if Ai vi = A0 v0 + · · · + Ai−1 vi−1 + Ai+1 vi+1 · · · + Am vm , for some v0 , v1 , . . , vm , then, by multiplying by Ai and keeping in mind 2. , we have Ai vi = 0.
N. Since the numbers xk are pairwise distinct, the polynomials x − xk are pairwise coprime. So the hypotheses of the Chinese remainder theorem are satisﬁed, with mk (x) = x − xk and the polynomials uk (x) = uk of degree zero (constants). 6) holds, and its uniqueness modulo the product n i=0 (x − xk ). Since this product has degree n + 1, u(x) is the only polynomial of degree at most n (or, possibly, the zero polynomial) that satisﬁes the conditions we required. It is called the Lagrange (interpolating) polynomial.
A solution for n = 4, that is, mod 24 , can be found as follows. With e3 = 1 we have: (1 + 22 c)2 = 1 + 23 c + 24 c2 ≡ 41 mod 24 , 41 − 1 = 5 ≡ 1 mod 2, c≡ 8 so e4 = 1 + 1 · 22 = 5, and 52 = 25 ≡ 41 mod 24 indeed. We can sum up the above in the following theorem. 9. Let a be an odd number. Then: i) a is always a square modulo 2; ii) a is a square modulo 4 if and only if a ≡ 1 mod 4; iii) a is a square modulo 2n , n ≥ 3, if and only if a ≡ 1 mod 8. 3 Newton’s method As already mentioned, series expansions yielding better and better approximations of a number are particular cases of a general method.