In this problem set, we use the following notation. Suppose E and L are field extensions of F. LetEmbf(E, L) := {0 : E → L | 0 is an F-linear injective ring homomorphism}.By F-linear, we mean 0(c)= c for every c E F. 3. Suppose E is a splitting field of g(x) E F[x] \ F_over F. Suppose E = F[a] for some a. Prove that|Embf(E,E)| = number of distinct zeros of ma,F(x) in E,and deduce that | Embf(E, E)| < [E : F].

Let σ∈EmbFE,E. Given that E=Fα. That is α is algebraic over F and mα,Fx=a0+a1x+...+akxk is the…

Answered: 3. Suppose E is a splitting field of…

3. Suppose E is a splitting field of g(x) E F[x] \ F over F. Suppose E = F[a] for some a. Prove that |Embf(E, E)| = number of distinct zeros of maf(x) in E, and deduce that | Embf(E,E)I < [E : F].

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

In this problem set, we use the following notation. Suppose E and L are field extensions of F. Let
Embf(E, L) := {0 : E → L | 0 is an F-linear injective ring homomorphism}.
By F-linear, we mean 0(c)
= c for every c E F.

3. Suppose E is a splitting field of g(x) E F[x] \ F_over F. Suppose E = F[a] for some a. Prove that
|Embf(E,E)| = number of distinct zeros of ma,F(x) in E,
and deduce that | Embf(E, E)| < [E : F].

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