Geometric Constructible Numbers

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Geometric constructible numbers are defined by the property that if 'a' is constructible, then the degree of the field extension [Q(a):Q] equals 2^n. However, the inverse is not universally true, as there exist algebraic numbers of order a power of 2 that are not constructible. Participants in the discussion express difficulty in finding clear counterexamples to illustrate this point. The conversation highlights the complexity of identifying such numbers, indicating that it may require extensive contemplation. Ultimately, the quest for a definitive counterexample remains unresolved.
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Geometric Constructible Numbers...

Hi, everyone.
I have a question about geometric constructible numbers.
I know that "if 'a' is constructible then [Q(a):Q]=2^n."
But I heard that its inverse is not true.
I want some counter examples about the inverse statement.
(I have checked by googling 'i' is a constructible number.)

Help...
 
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You mean you want a number that is algebraic of order a power of 2 and is NOT a constructible number? Hmm, now thats' a good question! I will need to think about that- for a few decades.
 


Yes, I mean that.
So, there was no clear answer about that?
 
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