I need to answer number 10,but I do not know how to show the answer. Please help

Question

I need to answer number 10,but I do not know how to show the answer.

Please help me. Thank you.

18 Introduction 10. (continued from section 1.2, Problem 12) As before, let Q[x] be the domain o polynomials over variable. Formulate definitions similar to those in the ext for prime and composite polynomials and for divisibility. Extend the results of Problems 1 and 9 above to the present context. (Continued in Section 1.4, Problem 8 11. a) Let n be a positive integer and let a1 a2, an 1 be any real numbers in the interval [0, 1) (that is, the interval 0 s a 1). Show that indices i and j exist such that i j and la al 1In. Hint: Use the informal version of Dirichlet's principle. What are the set and the subsets in the formal version, in this case?] b) Show that if a is real and qo qn are distinct positive integers, then there exist an integer r and indices i and j with 0 s i j s n such that la(qu r 1In c) show that if a is real and n is a positive integer, then there are integers q and r with 0 q s n and laa rl 1/n 1.4 REPRESENTATION SYSTEMS FOR THE INTEGERS We conclude this review of "elementary arithmetic from an advanced standpoint" with a discussion of several methods of representing the integers. So far we have an intrinsic description of the entire set z, as a domain with certain properties, and we have the descriptions 1, 1 1, 1 1 1, of the positive integers, which are useful for a definition but little more, at least out beyond 20 or so. A much more efficient system is the one everyone learns in elementary school, the decimal representation; it is so useful that every educated person has devoted five or more years to becoming familiar with it and learning to use the associated algorithms (carrying, rational operations. borrowing, long division, for performing the four just Let develop a slight generalization of it, by thinking about how works. The representation of an integer less than 1000, say, is determined by partitioning the interval [0, 1000) (containing the integers 0, 1 999) into 10 r equal parts [0, 100), [100, 200), [900, 1000), finding which the P integer belongs to, partitioning that subinterval into equal parts, and continuing until the integer is uniquely identified. If, for example, n e [8. 100, 9.100), and p. then n 100 e 10, and then in turn n 8.100 5. 10 E [7, 8), are the decimal representation of n is 857. Thus the 1000 integers from 0 to 999 partitioned into classes, each of which is partitioned into 10 Put s again into 10 each this way, the generalization is obvious. each containing a unique integer than 1 (in the Let m Part each mu 0), m, be integers larger Su and define on the interval [0, Mn to the m, subintervals [0, M. to 1)M. So MA)

Explanation / Answer

Prime Polynomial :A polynomial p(x) Q[x] is prime over Q if its degree is at least 1 and it cannot be factorised into polynomials of lower degree.

Composite Polynomial: All other polynomials which are not prime are called composite over Q.

Let f(x), g(x) be polynomials in Q[x]. We say that g(x) divides f(x) in Q[x] if f(x) = g(x)q(x) for some q(x) Q[x] (i.e. if f(x) is a multiple of g(x) in Q[x]).

(Division Algorithm in Q[x]). Let f(x) and g(x) be non-zero polynomials in Q[x]. Then there exist unique polynomials q(x) and r(x) in Q[x] with r(x) = 0 or deg(r(x)) < deg(g(x)) and f(x) = g(x)q(x) + r(x) .

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