Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 271nently impaired. Tartaglia sent Fior a list of 30 various mathematical problems;Fior countered by sending Tartaglia a list of 30 depressed cubics. Tartaglia wouldeither solve all 30 of the problems or absolutely fail. After much effort Tartagliafinally succeeded in solving the depressed cubic and defeated Fior, who faded intoobscurity.At this point another mathematician, Gerolamo Cardano (1501–1576), enteredthe story. Cardano wrote to Tartaglia, begging him for the solution to the depressedcubic. Tartaglia refused several of his requests, then finally revealed the solution toCardano after the latter swore an oath not to publish the secret or to pass it on toanyone else. Using the knowledge that he had obtained from Tartaglia, Cardanoeventually solved the general cubicax 3 + bx 2 + cx + d = 0.Cardano shared the secret with his student, Ludovico Ferrari (1522–1565), whosolved the general quartic equation,ax 4 + bx 3 + cx 2 + dx + e = 0.In 1543, Cardano and Ferrari examined del Ferro’s papers and discovered that hehad also solved the depressed cubic. Cardano felt that this relieved him of hisobligation to Tartaglia, so he proceeded to publish the solutions in Ars Magna(1545), in which he gave credit to del Ferro for solving the special case of the cubic.This resulted in a bitter dispute between Cardano and Tartaglia, who publishedthe story of the oath a year later.Exercises1. List all of the polynomials of degree 3 or less in Z 2 [x].2. Compute each of the following.(a) (5x 2 + 3x − 4) + (4x 2 − x + 9) in Z 12(b) (5x 2 + 3x − 4)(4x 2 − x + 9) in Z 12(c) (7x 3 + 3x 2 − x) + (6x 2 − 8x + 4) in Z 9(d) (3x 2 + 2x − 4) + (4x 2 + 2) in Z 5(e) (3x 2 + 2x − 4)(4x 2 + 2) in Z 5(f) (5x 2 + 3x − 2) 2 in Z 123. Use the division algorithm to find q(x) and r(x) such that a(x) = q(x)b(x) +r(x) with deg r(x) < deg b(x) for each of the following pairs of polynomials.(a) p(x) = 5x 3 + 6x 2 − 3x + 4 and q(x) = x − 2 in Z 7 [x](b) p(x) = 6x 4 − 2x 3 + x 2 − 3x + 1 and q(x) = x 2 + x − 2 in Z 7 [x]