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# sol6 - U.C Berkeley CS70 Discrete Mathematics and...

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U.C. Berkeley — CS70: Discrete Mathematics and Probability Problem Set 6 Lecturers: Umesh Vazirani & Christos H. Papadimitriou Due October 6, 2006 at 4:00pm Problem Set 6 Solutions 1. Polynomials with no roots A polynomial p ( x ) of degree n over a field F has at most n roots. But it does not need to have n roots, nor it must have roots at all. (a) Write all polynomials in GF 2 having no roots over GF 2 . (b) Write all polynomials in GF 3 having no roots over GF 3 . (c) Find a polynomial p ( x ) which has roots over GF 3 , but not over Z . Solution : Recall that polynomials of degree q in GF q can be rewritten as polynomials of degree q as x q = 1 for all x GF q . (a) The constant 1 polynomial is the only polynomial without roots. All non-constant polynomials have degree 1 and hence have one root. (b) These can be found by iterating through all 3 3 polynomials, but it is not necessary. First, notice that all non-zero constant polynomials have no roots and that all degree 1 polynomials have roots. Then, we are left with the 2 * 3 * 3 = 18 polynomials of degree 2. Here, observe that those with 0 constant term will have 0 as a root and exclude them. Now, you have 2 * 3 * 2 = 12 polynomials to consider. Moreover, you can arrange these in pairs

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