Geometric Constructible Numbers

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SUMMARY

The discussion centers on geometric constructible numbers, specifically the relationship between constructible numbers and their algebraic order. It is established that if 'a' is a constructible number, 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. The participants express a need for counterexamples to illustrate this point, indicating a gap in readily available information on the topic.

PREREQUISITES
  • Understanding of field extensions in algebra
  • Familiarity with constructible numbers in geometry
  • Knowledge of algebraic numbers and their properties
  • Basic concepts of Galois theory
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  • Research counterexamples of algebraic numbers that are not constructible
  • Study Galois theory and its implications for constructibility
  • Explore the properties of transcendental numbers versus algebraic numbers
  • Investigate the historical context of geometric constructions and their limitations
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Mathematicians, students of abstract algebra, and anyone interested in the foundations of geometry and number theory.

emptyboat
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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...
 
Last edited:
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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?
 
Last edited:

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