Polynomials and Irreducibility Proof with Prime Numbers and Complex Numbers, Assignments of Abstract Algebra

Download Polynomials and Irreducibility Proof with Prime Numbers and Complex Numbers and more Assignments Abstract Algebra in PDF only on Docsity! Assignment #1 Date Due: Thursday, Sept. 10, 2009 Section 0 numbers 3, 5, 9, 16, 18 and a) Let R[x] denote the set of all polynomials in the variable x with real number coe!- cients, i.e. R[x] = ! anx n + an!1x n!1 + · · · + a0 | ai ! R and n ! N " Ex: 5x2 " 17" 10 x + 2 and !x107 " 15 are polynomials (sums of real numbers times non- negative interger powers of x). Prove that every polynomial in R[x] of positive degree is either irreducible itself or is a product of irreducible polynomials in R[x]. (An irreducible polynomial h(x) is one that cannot be factored into a product of polynomials of strictly lower degree.) b) Let p be a prime integer: i) Prove that one cannot find nonzero positive integers a and b sucht that a2 = pb2. Hint: Use a proof by contradiction: suppose there are such intergers a and b. Explain why one may assume a and b to be relatively prime. (Recall: two positive integers are relatively prime if they have no common factors other than 1). Then proceed to show that p|a and p|b. ii) Deduce from (i) that # p is irrational, i.e., # p $! Q, if p is prime. c) Compute each of the following: i) i27 ii) (1" i)(2 + i) (1" 2i)(1 + i) iii) |3" 4i| iv) # i 0 1 "i $4 v) # i 0 1 "i $!1 d) Express the following in polar-coordinate form z = |z|(cos " + i sin "). i) z = i ii) z = 7 iii) z = 1 + # 3 i