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

If you want to continue the sequence indefinitely, you can write it as \sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{7}+...} \in F_n, where F_n represents an infinite tower of extension fields. In summary, we are looking for a sequence of extension fields Q= F_0\subseteq...\subseteq F_n, where \sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}} \in F_n, and all steps are non-trivial except for the last one. This sequence can continue infinitely, with each step adding a new square root.
  • #1
saadsarfraz
86
1

Homework Statement



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

where [tex]\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}[/tex] [tex]\in[/tex] F[tex]_{n}[/tex]

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.[tex]\sqrt{}[/tex]
 
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  • #2
The "trivial" step is [itex]F_1= Q(\sqrt{1})[/itex] since [itex]\sqrt{1}= 1[/itex] which already is a rational number. Take [itex]F_2= F_1(\sqrt{2})= Q_(\sqrt{2})[/itex], [itex]F_3= F_2(\sqrt{3})[/itex], etc.
 
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  • #3
ok i think i got it, can anyone please check my answer

K_0 = 2 which corresponds to F_0
K_1= 1 + [tex]\sqrt{2}[/tex] for F_1
K_2= 1 + [tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] for F_2
K_3= 1 + [tex]\sqrt{2}[/tex] + [tex]\sqrt{3}[/tex] + [tex]\sqrt{5}[/tex] for F_3
K_4= [tex]\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}[/tex] for F_4except the last one is supposed to go on forever? can anyone help me in this.
 
  • #4
Your original question did not "go on forever", it stopped at [itex]\sqrt{5}[/itex].
 

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

1. What is a sequence of extension fields?

A sequence of extension fields is a series of fields that are created by successively adding elements to a base field. Each field in the sequence contains the previous field and at least one new element. This is often used in algebraic number theory to construct fields with specific properties.

2. How can a sequence of extension fields be used to prove a value?

In the case of proving a value like $\sqrt{1+\sqrt{2}+\sqrt{3}+\sqrt{5}}$, a sequence of extension fields can be used to construct a field that contains this value. By starting with a base field and adding successive square roots, we can eventually reach a field that contains all of the necessary elements to prove the value.

3. Why is it important to use extension fields in the proof?

Using extension fields allows us to construct a field that contains the value we are trying to prove. This is important because it allows us to manipulate the elements in the field and use algebraic techniques to prove the value. Without extension fields, it may be difficult or impossible to prove certain values.

4. What is the significance of the square roots in this particular sequence of extension fields?

The square roots in this sequence have specific properties that allow us to construct a field with the desired value. For example, the square root of 2 is used because it is irrational and therefore cannot be expressed as a ratio of two integers. This property is important in constructing fields with unique and specific properties.

5. Can a sequence of extension fields be used to prove other values?

Yes, a sequence of extension fields can be used to prove a variety of values in algebraic number theory. This method is particularly useful for proving values that involve square roots or other irrational numbers. By carefully selecting the elements in the sequence, we can construct fields that contain these values and use them in our proofs.

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