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I have the following idea:
If a field F is algebraic over a field K, then F is a finite extension of K.
Is this true?
I was trying to come up with a counterexample. Something like F = Q([tex]\sqrt{p}[/tex] , for all primes p). Q are the rationals, of course.
Then F wouldn't be a finite extension of Q, but I don't know if every element in F is algebraic over Q.
I would appreciate any comments.
If a field F is algebraic over a field K, then F is a finite extension of K.
Is this true?
I was trying to come up with a counterexample. Something like F = Q([tex]\sqrt{p}[/tex] , for all primes p). Q are the rationals, of course.
Then F wouldn't be a finite extension of Q, but I don't know if every element in F is algebraic over Q.
I would appreciate any comments.