Question

To complete the proof of Theorem 3.4.6, we need to show that a basis for the extension field F:K is αᵢ βⱼ , where αᵢ is from the basis for E:K, and βⱼ is from the basis for F:E.

Proof:

1. Let's first establish that {αᵢ βⱼ} spans F:K:

Since {αᵢ} is a basis for E:K, and {βⱼ} is a basis for F:E, any element in E can be expressed as a linear combination of αᵢ, and any element in F can be expressed as a linear combination of βⱼ. Therefore, any element in F can be expressed as a linear combination of both αᵢ and βⱼ, which means it can be expressed as a linear combination of αᵢ βⱼ. This shows that {αᵢ βⱼ} spans F:K.

Answer 1

In the proof of Theorem 3.4.6, we establish that {αᵢ βⱼ} spans the extension field F:K.

Step-by-step explanation:

In the proof of Theorem 3.4.6, we need to show that a basis for the extension field F:K is αᵢ βⱼ , where αᵢ is from the basis for E:K, and βⱼ is from the basis for F:E.

To prove this, we first establish that {αᵢ βⱼ} spans F:K. Since {αᵢ} is a basis for E:K, and {βⱼ} is a basis for F:E, any element in E can be expressed as a linear combination of αᵢ, and any element in F can be expressed as a linear combination of βⱼ. Therefore, any element in F can be expressed as a linear combination of both αᵢ and βⱼ, which means it can be expressed as a linear combination of αᵢ βⱼ. This shows that {αᵢ βⱼ} spans F:K.