isider the complex number 3 a = 4+i√2, where i² = -1. (a) Find the minimal polynomial of a over Q. (Don't forget to prove that this polynomial is irreducible!) (b) Compute the degree [Q(a): Q] and find a basis of Q(a) over Q consisting of powers of a. (c) Write a as a Q-linear combination of the basis elements from part (b)
isider the complex number 3 a = 4+i√2, where i² = -1. (a) Find the minimal polynomial of a over Q. (Don't forget to prove that this polynomial is irreducible!) (b) Compute the degree [Q(a): Q] and find a basis of Q(a) over Q consisting of powers of a. (c) Write a as a Q-linear combination of the basis elements from part (b)
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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![Consider the complex number
= √√4 + i√2,
α =
where i² = -1.
(a) Find the minimal polynomial of a over Q. (Don't forget to prove that this polynomial is
irreducible!)
(b) Compute the degree [Q(a): Q] and find a basis of Q(a) over Q consisting of powers of a.
(c) Write a as a Q-linear combination of the basis elements from part (b)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed5788d2-a0b4-4eb1-998c-a4248e13736e%2F0b256bcc-e04c-42b6-865a-86a6a7a6da65%2Flijvj4_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the complex number
= √√4 + i√2,
α =
where i² = -1.
(a) Find the minimal polynomial of a over Q. (Don't forget to prove that this polynomial is
irreducible!)
(b) Compute the degree [Q(a): Q] and find a basis of Q(a) over Q consisting of powers of a.
(c) Write a as a Q-linear combination of the basis elements from part (b)
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