Let a € Zz be a zero of the irreducible polynomial p(x) = x3 + x² + 2 in Z3[x]. a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of an element in Z3(a). b. Show that 2a2 +1 is also a zero of p(x). c. Explain why deg(8, Z3) = 3 for all BE Z3(a) \ Zg.
Let a € Zz be a zero of the irreducible polynomial p(x) = x3 + x² + 2 in Z3[x]. a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of an element in Z3(a). b. Show that 2a2 +1 is also a zero of p(x). c. Explain why deg(8, Z3) = 3 for all BE Z3(a) \ Zg.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let a e Zz be a zero of the irreducible polynomial p(x) = x³ + x² + 2 in Z3[x].
a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of an
element in Z3(a).
b. Show that 2a2 +1 is also a zero of p(r).
= 3 for all BE Z3(a) \ Z3.
c. Explain why deg(B, Z3)
d. Determine irr(2a, Z3).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18c51d48-2201-40b6-bcbd-9606bca209af%2Fc3490071-ffec-4b1d-b6dd-ad40a5957e5d%2F14ezhyo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a e Zz be a zero of the irreducible polynomial p(x) = x³ + x² + 2 in Z3[x].
a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of an
element in Z3(a).
b. Show that 2a2 +1 is also a zero of p(r).
= 3 for all BE Z3(a) \ Z3.
c. Explain why deg(B, Z3)
d. Determine irr(2a, Z3).
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