# Number fields

1. Apr 1, 2009

1. The problem statement, all variables and given/known data

Find a sequence of extension fields (i.e. tower)
Q= F$$_{0}$$$$\subseteq$$.......$$\subseteq$$F$$_{n}$$.

where $$\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}$$ $$\in$$ F$$_{n}$$

Prove that all the steps are non-trivial. except the last one. btw Q is the set of rational number. and 0 and n on F were meant to be subscripts not superscripts (i dont know how to do that)

2. Relevant equations

3. The attempt at a solution

I'm a bit confused as to what to do in this question? I dont think I understand the question.$$\sqrt{}$$

Last edited: Apr 1, 2009
2. Apr 1, 2009

### HallsofIvy

Staff Emeritus
The "trivial" step is $F_1= Q(\sqrt{1})$ since $\sqrt{1}= 1$ which already is a rational number. Take $F_2= F_1(\sqrt{2})= Q_(\sqrt{2})$, $F_3= F_2(\sqrt{3})$, etc.

Last edited: Apr 3, 2009
3. Apr 2, 2009

ok i think i got it, can anyone please check my answer

K_0 = 2 which corresponds to F_0
K_1= 1 + $$\sqrt{2}$$ for F_1
K_2= 1 + $$\sqrt{2}$$ + $$\sqrt{3}$$ for F_2
K_3= 1 + $$\sqrt{2}$$ + $$\sqrt{3}$$ + $$\sqrt{5}$$ for F_3
K_4= $$\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}$$ for F_4

except the last one is supposed to go on forever? can anyone help me in this.

4. Apr 3, 2009

### HallsofIvy

Staff Emeritus
Your original question did not "go on forever", it stopped at $\sqrt{5}$.