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Reading Questions 21.4 Reading Questions

1.

What does it mean for an extension field \(E\) of a field \(F\) to be a simple extension of \(F\text{?}\)

2.

What is the definition of a minimal polynomial of an element \(\alpha\in E\text{,}\) where \(E\) is an extension of \(F\text{,}\) and \(\alpha\) is algebraic over \(F\text{?}\)

3.

Describe how linear algebra enters into this chapter. What critical result relies on a proof that is almost entirely linear algebra?

4.

What is the definition of an algebraically closed field?

5.

What is a splitting field of a polynomial \(p(x)\in F[x]\text{?}\)