I'm suppossed to use an example to show the following statement. If F over K is galois but not algebraic and L is an intermediate field between K and F, then F over L is not galois. Any help at all would be greatly appreciated.
field extension problem
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field-theory
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4The most common definition of "Galois extension" requires that the extension be algebraic. What definition are you using? – 2011-03-28
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0@ Brad I think that F must have char 0 – 2011-03-28
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0@user8771, a Galois extension may well be infinite but it still needs to be algebraic under the common definition. Also, if F is an extension of K and L is an intermediate field, then "K over L" doesn't make sense. – 2011-03-28
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0Thank you..I just edited it. – 2011-03-28
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1Ok, that still leaves the question of what "Galois but not algebraic" means. Whatever the definition, the question seems odd - F/L is not Galois for all L except L=K? – 2011-03-28
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3@All Hungerford's Algebra book defines a Galois extension as an extension $F/K$ such that the fixed field of $\text{Aut}_K F$ is $K$, without requiring it to be algebraic, although he explicitly notes that it is commonly required to be algebraic. – 2011-03-28