By Antonio Machì (auth.)

This e-book bargains with a number of themes in algebra beneficial for laptop technology purposes and the symbolic therapy of algebraic difficulties, mentioning and discussing their algorithmic nature. the subjects lined variety from classical effects comparable 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, in particular through Berlekamp’s strategy, and the discrete Fourier rework. uncomplicated algebra innovations are revised in a kind suited to implementation on a working laptop or computer algebra system.

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48 2 p-adic series expansions Let us expand now 13 mod 5. 31. 00 . . = 0. 2. It is quite helpful to see how to expand 13 as a series of powers of 15 . 13. 3 5 5 5 5 So we see that the period is inverted with respect to − 13 , and starts after the decimal point, that is, with the positive powers of 15 . Moreover, before the point we have 0. It is easy to see that this is a general fact. 3 3. Let us see now an example with p = 10. Expand − 11 . We have 102 ≡ 2 1 mod 11, so the period is d = 2. Moreover, 10 − 1 = 11 · 9, so m = at = 3 · 9 = 27.

With x = p, or x = pn , we obtain the above expressions, as well as, with x = a, a = 1, every expression 1 . We shall come back to this fact when we shall discuss series expansions of 1−an rational functions. Let us see now a standard theorem. 5. A p-adic expansion represents a rational number if and only if it is periodic. Proof. c1 c2 . . cd−1 , or: x = c0 + c1 p + · · · + cd−1 pd−1 + c0 pd + · · · + cd−1 p2d−1 + · · · . Factoring pd out, this expression can be written as: x = c0 + c1 p + · · · + cd−1 pd−1 + (c0 + c1 p + · · · + cd−1 pd−1 )pd + · · · , that is: x = (c0 + c1 p + · · · + cd−1 pd−1 )(1 + pd + p2d + · · · ), and keeping in mind that the right-hand series sums to 1/(1 − pd ), x = c0 + c1 p + · · · + cd−1 pd−1 · 1 c0 + c1 p + · · · + cd−1 pd−1 = .

The existence and uniqueness of such a polynomial u(x) can be seen using the theory of systems of linear equations. For the unknown polynomial u(x) = n n i i i=0 ai x , the relations i=0 ai xk = uk , k = 0, 1, . . , n have to hold. So we have a system of n + 1 equations in the n + 1 unknowns a0 , a1 , . . , an , whose determinant is the Vandermonde polynomial V of xi . Since the xi are pairwise distinct, we have V = 0, so the solution exists and is unique. As in the case of integers, we have two methods to compute u(x).