I EXAMPLE 8 The Splitting Field of x" – a over Q Let a be a positive rational number and let w be a primitive nth root of unity (see Example 2 in Chapter 16). Then each of all", wa", w²al", ..., o"-'ah is a zero of x" – a in Q(Va, w).

I EXAMPLE 8 The Splitting Field of x" – a over Q Let a be a positive rational number and let w be a primitive nth root of unity (see Example 2 in Chapter 16). Then each of all", wa", w²al", ..., o"-'ah is a zero of x" – a in Q(Va, w).

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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Question

Let F(a) be the field described in Exercise 8. Show that a² and a² + a are zeros of x³ + x + 1.

I EXAMPLE 8 The Splitting Field of x" – a over Q
Let a be a positive rational number and let w be a primitive nth root of
unity (see Example 2 in Chapter 16). Then each of
all", wa", w²al", ..., o"-'ah
is a zero of x" – a in Q(Va, w).

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