By Antonio Machì (auth.)
This booklet offers with numerous themes in algebra invaluable for computing device technology purposes and the symbolic therapy of algebraic difficulties, declaring and discussing their algorithmic nature. the subjects lined diversity from classical effects resembling 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, specially through Berlekamp’s technique, and the discrete Fourier rework. easy algebra strategies are revised in a kind fitted to implementation on a working laptop or computer algebra system.
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Extra resources for Algebra for Symbolic Computation
Let us see now the case of a quadratic equation, x2 − a = 0, with p prime and p not dividing a (that is, a is not zero modulo p). Let us consider now only the case p = 2. Note ﬁrst that, unlike the linear case, this equation might not have solutions even for n = 1. It is, for instance, the case of x2 − 2 = 0 with p = 3. But if there is a solution mod p, then there is one mod pn for every n. Denote this √ solution by a. 6. Let p be prime and p > 2. If the equation x2 − a = 0, p a, has a solution in integers modulo p, then it has one in integers modulo pn for all n.
3 Polynomials 25 Finally, with m2 (x) = x2 − 2, m0 (x)m1 (x) = x3 − x2 + x − 1 we have − 2 + x2 x+1 · (x2 − 2) + · (x3 − x2 + x − 1) = 1 3 3 1 3 2 4 and L2 (x) = x+1 3 · (x − x + x − 1) = 3 (x − 1). It is immediate to verify that L0 (x) + L1 (x) + L2 (x) = 1, as well as the other two properties of the Lk (x)s. Let x3 − 1 be a polynomial in A; we have: x3 − 1 = (x − 1)q0 (x) + 0, x3 − 1 = (x2 + 1)q1 (x) + (−x − 1), x3 − 1 = (x2 − 2)q2 (x) + 2x − 1. It follows that x3 − 1 ≡ 0 · L0 (x) + (−x − 1)L1 (x) + (2x − 1)L2 (x) mod m(x).
Conversely, if x is negative, −x is positive, so −x = d0 , d1 d2 . . 0, and hence x = −d0 − d1 p − · · · − dk pk + 0 · pk+1 + · · · . In the usual reduction we have: · · ·+(p−dk −1)pk −1·pk+1 +0·pk+2 +· · · = · · · (p−1)pk+1 −1·pk+2 +· · · . So we notice that all the coeﬃcients of the powers pi with i > k are equal to p − 1. ♦ Examples. 1. Determine the p-adic expansion of 1 . 1 we have: 1 · (1 − p) + 1 · p = 1, and hence: 1 1 =1+ p, 1−p 1−p and c0 = 1. From this follows 1 1 1 2 = 1 + (1 + p)p = 1 + p + p , 1−p 1−p 1−p and c1 = 1.
Algebra for Symbolic Computation by Antonio Machì (auth.)