Algebraic And Simple Extensions

[WannaBe22]

Homework Statement

I'll be delighted to get an answer to the following question:
Does every algebraic extension of a field is a simple extension?

Homework Equations

The Attempt at a Solution

I'm pretty sure that the answer is negative... I was thinking on taking the field of all the algebraic numbers over Q ... this field is obviously an algebraic extension of Q, but how can I prove it isn't simple? (I'm pretty sure that its degree is infinity, but have no idea how to to prove it) ...

Hope you'll be able to help me

Thanks!

 

[Office_Shredder]

To prove the degree you can use the tower law. 2^{1/n} is obviously in the algebraic numbers for each n, so if A is the set of algebraic numbers

[A:Q]=[A:Q(2^{1/n})][Q(2^{1/n}):Q]

 

[WannaBe22]

Thanks a lot!