Prove the impossibility of construction with ruler and compass the numbers:

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SUMMARY

The discussion centers on proving the impossibility of constructing certain numbers using a ruler and compass, specifically ∛4, ∛5, ∛3, and ∛a. Participants express challenges in approaching the problem and share their initial attempts at solutions. The conversation highlights the need for a deeper understanding of field extensions and constructible numbers in the context of classical geometry.

PREREQUISITES
  • Understanding of constructible numbers in geometry
  • Familiarity with field extensions in algebra
  • Knowledge of the properties of cubic roots
  • Basic skills in geometric constructions using a ruler and compass
NEXT STEPS
  • Research the criteria for constructible numbers in classical geometry
  • Study field extensions and their implications for geometric constructions
  • Explore the properties of cubic roots and their algebraic representations
  • Learn about the historical context and proofs related to ruler and compass constructions
USEFUL FOR

Mathematics students, educators, and enthusiasts interested in geometric constructions and algebraic proofs, particularly those focusing on the limitations of classical methods.

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Homework Statement



∛4 ∛5 ∛3 and ∛a

Homework Equations





The Attempt at a Solution


I have started with the problem but i feel like I'm off track. I'll upload a pic of the work so far.
 
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here is that I've started on
 

Attachments

  • thing.jpg
    thing.jpg
    23 KB · Views: 447

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