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.