By F. M. Hall

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A/. a/ D 0 if, and only if, x a divides p. Proof. x a/Cr, where the remainder r is a constant. a/ D r. 23. ˛/ D 0. Say the multiplicity of the root ˛ is k if x ˛ appears exactly k times in the irreducible factorization of p. 24. A polynomial p 2 KŒx of degree n has at most n roots in K, counting with multiplicities. That is, the sum of multiplicities of all roots is at most n. Proof. p/. 2. 1.

X/ D 3 6 x C 4 x 2 Repeated division with remainder gives . 4 C 9x 3 x2 6 x3 C 6 x4 D . 3 6 x C 4 x2 2 x3 C x4: 3 x5 C x6/ 6 x C 4 x2 2 x 3 C x 4 / C . 4 C 12 x 12 x 2 C 4 x 3 /; 2 x3 C x4 / x 1 C /. 4 C 12 x 12 x 2 C 4 x 3 / C . 4 8 x C 4 x 2 /; 4 4 . 4 C 12 x 12 x 2 C 4 x 3 / D . 1 C x /. 4 8 x C 4 x 2 / C 0: D. x/ as follows: The sequence of quotients produced in the algorithm is q1 D x C xÄ2 , q2 D x4 C 14 , and q3 D 1 C x . The qk determine matrices 0 1 Qk D . x/ comprise the first column 1 qk of the product Q D Q1 Q2 Q3 .

Conclude that any permutation can be written as a product of some number of 2-cycles. 3 to discover the right pattern. Then do a proper proof by induction. 6. Explain how to compute the inverse of a permutation that is given in two-line notation. 7. Explain how to compute the inverse of a permutation that is given as a product of cycles (disjoint or not). One trick of problem solving is to simplify the problem by considering special cases. First you should consider the case of a single cycle, and it will probably be helpful to begin with a short cycle.