Primitive 5th root of unity extension

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SUMMARY

The discussion focuses on the field extension E = Q(a), where a is a primitive fifth root of unity. The basis for E as a vector space over Q is established as {1, a, a^2, a^3}. The user seeks assistance in expressing the ratio a/(a-3) in terms of this basis, indicating a need to apply concepts from both Galois theory and analysis to solve the problem effectively.

PREREQUISITES
  • Understanding of field extensions in algebra
  • Familiarity with primitive roots of unity
  • Knowledge of vector spaces over Q
  • Basic concepts of Galois theory
NEXT STEPS
  • Research the properties of primitive roots of unity in field extensions
  • Learn how to express elements in terms of a basis in vector spaces
  • Study the application of analysis in Galois theory problems
  • Explore the implications of the ratio a/(a-3) in the context of field extensions
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Mathematicians, particularly those specializing in algebra and number theory, as well as students studying Galois theory and field extensions.

mrbohn1
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Hi,

Let E = Q(a), where a is a primitive fifth root of unity. Find a basis for E as a vector space over Q, and express a/a-3 in terms of this basis.

I can find a basis for E: {1, a, a^2, a^3}, but am not sure how to express a/a-3 in terms of this basis. Any help much appreciated.
 
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Hint: If |z|<1, do you know of another way of writing 1/(1-z)?
 
Last edited:
aaah...I see - thanks. I wasn't expecting to have to use analysis in a galois theory exercise...
 

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