Is Every Algebraic Element of a Field Extension Contained in the Base Field?

[futurebird]

My notes say:

If K is an extension of F then A={a in K | a is algebraic} is a subfield of K contained in F.​

But the elements in A need not be in F, right? Shouldn't it be:

If K is an extension of F then A={a in K | a is algebraic} is a subfield of K contained in K.​

But I don't see the point of saying that-- if it's a subfield, of course it's in K. Also, I thought that AUF = K ... but that only happens when it is an algebraic extension... right?

 

[Hurkyl]

If K is an extension of F then A={a in K | a is algebraic} is a subfield of K containing in F.​

I fixed a probably typo. And I assume you meant "a is algebraic over F"?

 

[futurebird]

A={a in K | a is algebraic over F}

Like this, yes.

 

[futurebird]

never mind--- I get it now!

Thanks again!