Proving a Sequence of Extension Fields for $\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}$

[saadsarfraz]

Homework Statement

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 don't know how to do that)

Homework Equations

The Attempt at a Solution

I'm a bit confused as to what to do in this question? I don't think I understand the question.$$\sqrt{}$$

 

[HallsofIvy]

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.

 

[saadsarfraz]

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_4except the last one is supposed to go on forever? can anyone help me in this.

 

[HallsofIvy]

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