By Antonio Machì (auth.)

This e-book bargains with numerous subject matters in algebra valuable for laptop technology functions and the symbolic therapy of algebraic difficulties, declaring and discussing their algorithmic nature. the subjects lined 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 services, to arrive the matter of the polynomial factorisation, particularly through Berlekamp’s procedure, and the discrete Fourier rework. easy algebra suggestions are revised in a sort suited to implementation on a working laptop or computer algebra system.

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**Example text**

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. 2. Remark. The usual decimal expression of a proper fraction ab with a denominator coprime with 10 can be obtained as above with p = 10. More precisely, having found the least d such that 10d ≡ 1 mod b, that is, the period, let 10d − 1 = bt. Then, as above, at m a = d = d .

Proof. First of all, note that if ∂a < ∂g, then necessarily ∂b < ∂f , otherwise ∂h = ∂(af + bg) = ∂bg ≥ ∂f + ∂g, against the hypothesis. Suppose now that ∂a ≥ ∂g; dividing we get a = gq + r with ∂r < ∂g; so, set a1 = a − gq and b1 = b + f q and we get a1 f + b1 g = h; since ∂a1 < ∂g we ﬁnd ∂b1 < ∂f , by the above argument. 5. Let f and g be two relatively prime polynomials. Then there exist two polynomials a and b with ∂a < ∂g and ∂b < ∂f , and such that af + bg = 1. Moreover, a and b are uniquely determined.

4 Series expansion of rational functions A polynomial f (x) with coeﬃcients in a ﬁeld (which in what follows will be the ﬁeld Q of rationals) is a linear combination of the monomials 1, x, x2 , . , with coeﬃcients equal to zero from some point on: f (x) = a0 + a1 x + · · · + an xn + 0 · xn+1 + · · · . In this form a polynomial is a (ﬁnite) series of powers of x, or in base x. If p(x) is an arbitrary polynomial of ﬁrst degree, we have analogously the expansion: f (x) = c0 + c1 p(x) + c2 p(x)2 + · · · + cn p(x)n + 0 · p(x)n+1 + · · · in base p(x).