# Algebraic And Simple Extensions

1. Sep 22, 2010

### WannaBe22

1. The problem statement, all variables and given/known data
I'll be delighted to get an answer to the following question:
Does every algebraic extension of a field is a simple extension?
2. Relevant equations
3. 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!

2. Sep 22, 2010

### Office_Shredder

Staff Emeritus
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

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

3. Sep 24, 2010

### WannaBe22

Thanks a lot!