Download Algorithms for Discrete Fourier Transform and Convolution by R. Tolimieri, Myoung An, Chao Lu (auth.), C. S. Burrus PDF

By R. Tolimieri, Myoung An, Chao Lu (auth.), C. S. Burrus (eds.)

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For any two polynomials f(x) and g(x) over Q[x], show that the following set is an ideal, J = {a(x)f(x) + b(x)g(x) I a(x),b(x) E Q[x]}. 19. Let F be a finite field and form the set L = {I, 1 + 1, ... '. + 1, ... } Chapter 1. Introduction to Abstract Algebra 35 Show that L has order p for some prime p and that L is a subfield of F isomorphic to the field Z / p. The prime p is called the characteristic of the finite field F. 20. Show that every finite field K has order pR for some prime p and integer n ~ 1.

Introduction to Abstract Algebra 33 7. Give the table for the ring-isomorphism ~ of the CRT corresponding to factorizations n = 4 X 5 and n = 2 X 7. 8. Suppose n = nl n2 ... , nr are relatively prime in pairs. Show that 1 ~ Ie ~ r. 9. Continuing the notation and using the result of problem 8, define integers el, e2, ... , er satisfying e1c == 0 mod n" 1 ~ 1e,1 < r, Ie =/: I. These integers are uniquely determined by these conditions and form the system of idempotents corresponding to the factorization given in Problem 8.

The unit group (11) is cyclic, but U(2a), a > 2, is never cyclic. The exact result follows. Theorem 3. The group (12) is the direct product of two cyclic groups, one of order 2, the other of order 2a - 2• In fact, if 0 1 O 2 = {Sic (13) = {1, -1}, I0~k < 2a - 2 }, (14) then (15) Example 5. For p = 3 and a = 2, Example 6. Take p = 2 and a = 3. Then U(23) = {1, 3, 5, 7} = {1, 7} x {1, 5}. 6. Polynomial Rings Take a = 4. Then {I, IS} X {I, 5, 9, 13}. 6. Polynomial Rings Consider a field F and denote by F[z] the ring of polynomials in the indeterminate z, having coefficients in F.

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