By Antonio Machì (auth.)
This e-book offers with a number of subject matters in algebra beneficial for machine technological know-how functions and the symbolic therapy of algebraic difficulties, declaring and discussing their algorithmic nature. the themes lined diversity from classical effects similar 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, specially through Berlekamp’s technique, and the discrete Fourier rework. uncomplicated algebra suggestions are revised in a kind fitted to implementation on a working laptop or computer algebra system.
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This e-book offers with numerous issues in algebra priceless for desktop technological know-how functions and the symbolic remedy of algebraic difficulties, declaring and discussing their algorithmic nature. the subjects lined diversity from classical effects akin 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.
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Extra info for Algebra for Symbolic Computation
Example. Consider the equation x2 − 41 = 0. We have the solution x = 1 for n = 1, 2, 3. 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.
5) we see that the value f (α) of a polynomial f (x) at a point α is the value at α of the remainder of the division of f (x) by x − α. This division yields a way to compute f (α) which is more eﬃcient that the usual method involving the computation of the powers of α. With the usual method we have to perform the n − 1 multiplications α · α = α2 , α2 · α = α3 ,. . , αn−1 · α = αn , and then the n multiplications an−i αi ; all in all, 2n − 1 multiplications. Note how we ﬁnd instead the coeﬃcients of the quotient q(x) from those of f (x): starting from a0 , we multiply by α and add the following coeﬃcient: a0 , a0 α + a1 , (a0 α + a1 )α + a2 , .
En , for all n: input: a, b, p, n; output: c0 , c1 , . . cn−1 (or: e1 , e2 , . . , en ); d := 1/b mod p; c0 := ad mod p; e1 := c0 ; for k from 1 to n − 1 do: qk := quotient(a − bek , pk ); ck := dqk mod p; ek+1 := ek + ck pk . The expansion of a non-negative integer number in base p is periodic, with period 1: the period consists of a single digit, zero. c1 . . ck ck+1 ck+2 . . ck+d , b where this expression means that ck+1 = ck+d+1 = ck+2d+1 = . , and analogously for the other ci s. If d is the least length for which we have these inequalities, then we say that a/b is periodic of period d.
Algebra for Symbolic Computation by Antonio Machì (auth.)