Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements. (b) Use the above problem to construct a field with 27 elements.

Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements. (b) Use the above problem to construct a field with 27 elements.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 17E: Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive...

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Question

Suppose f(r) E Z,[r] and f(x) is irreducible over Z,, where p is prime.
(a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements.
(b) Use the above problem to construct a field with 27 elements.

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