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Galois Theory : Show that any algebraic extension of a perfect field is perfect.

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Show that any algebraic extension of a perfect field is perfect (using the below hint only).

Hint: Let K be a perfect field and F an algebraic extension of K. If F is not perfect, then there is a polynomial f(x) an element of F[x] that has an irreducible factor p(x) with a repeated root u. Here u is algebraic over K; let g(x) be the minimal polynomial of u over K. What is the relationship between g(x) and p(x)? Show that g(x) has a repeated root - this contradicts the hypothesis that K is perfect.

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Algebraic extensions of perfect fields are investigated. The solution is detailed and well presented.

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Proof:
Let K be a perfect field and F an algebraic extension of K. If F is not perfect, then there is a polynomial f(x), an element of F[x], that has an irreducible factor p(x) with a repeated root u. Here u is algebraic over K; let g(x) be the ...

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