By Antonio Machì

This publication bargains with a number of themes in algebra helpful for desktop technological know-how purposes and the symbolic therapy of algebraic difficulties, declaring and discussing their algorithmic nature. the themes coated variety 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 capabilities, to arrive the matter of the polynomial factorisation, specially through Berlekamp’s procedure, and the discrete Fourier rework. simple algebra thoughts are revised in a kind fitted to implementation on a working laptop or computer algebra procedure.

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**Example text**

Mod 7. 12 So the expansion is periodic, with period 2 (the last two digit 6 and 2). 62 mod 7. 3 we have then that the equation 12x − 1 = 0 has solution 3 in integers mod 7, solution 3 + 6 · 7 = 45 in integers mod 72 (indeed, 12 · 45 − 1 = 540 − 1 = 539 = 11 · 72 ≡ 0 mod 72 ), and so on. So the procedure is pretty analogous to the one for integers. More precisely, from ab = c0 + db p we have: a b hence d = a−bc0 p . − c0 1 a − bc0 d = = , p b p b Now, d = bc1 + pd1 ≡ bc1 mod p, so: c1 ≡ db−1 = Analogously, d1 = a−b(c0 +c1 p) p2 c2 = 1 a − bc0 mod p.

If a/b is an integer, that is if b = 1, then the order of p mod 1 is 1. Indeed, since two arbitrary integers are always congruent mod 1, the least positive integer d such that pd ≡ 1 mod b is 1. So we have found again the fact that the period of an integer is 1. 2. The negative rational numbers ab , a < b, are always represented by purely periodic numbers. For the positive ones, there are always further digits before the period. Examples. 1. Let us expand − 13 mod 5. We have 52 ≡ 1 mod 3, so the period is d = 2.

Xn nor on uk+1 , . . , un . The computation of the ai s is done as in the case of integers: u0 = u(x0 ) = a0 , u1 = u(x1 ) = a0 + a1 (x1 − x0 ), and hence: a1 = u1 − u0 . x1 − x0 Moreover, u2 = u(x2 ) = a0 + a1 (x2 − x0 ) + a2 (x2 − x0 )(x2 − x1 ) u1 − u0 = u0 + (x2 − x0 ) + a2 (x2 − x0 )(x2 − x1 ), x1 − x0 32 1 The Euclidean algorithm, the Chinese remainder theorem so: a2 = (u2 − u0 )(x1 − x0 ) − (u1 − u0 )(x2 − x0 ) , (x1 − x0 )(x2 − x0 )(x2 − x1 ) and so on. 8), but they are actually the same formula.