Problem 3. Let F ⊆ E be a field extension. (i) Suppose α ∈ E is...

Problem 3. Let F ⊆ E be a field extension.

(i) Suppose α ∈ E is algebraic of odd degree over F. Prove that F(α) = F(α^2 ). Hints: look at the tower of extensions F ⊆ F(α^2 ) ⊆ F(α) and their degrees.

(ii) Let S be a (possibly infinite) subset of E. Assume that every element of S is algebraic over F. Prove that F(S) = F[S]

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Colby Messinger answered 1 year ago

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