Showing Gal(E/Q) is Isomorphic to Z4

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Discussion Overview

The discussion revolves around demonstrating that the Galois group Gal(E/Q) is isomorphic to Z4, where E is the splitting field for the polynomial X^5-1 over the rational numbers Q. The scope includes theoretical aspects of Galois theory and the properties of the roots of the polynomial.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant seeks assistance in showing that Gal(E/Q) is isomorphic to Z4.
  • Another participant identifies the roots of the polynomial X^5-1, noting the single real root and the four complex roots, and questions how to construct the Galois group from this information.
  • A third participant describes Z4 as having elements {0,1,2,3} and mentions that only two elements have an order of 4, expressing uncertainty about further requirements to establish isomorphism.
  • A subsequent reply challenges the claim about the orders of elements in Z4 and prompts a discussion about the relationships between the non-trivial roots of X^5-1.
  • One participant concludes that they have resolved their understanding and only need to demonstrate that the Galois group is cyclic.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the properties of Z4 and the Galois group, indicating that the discussion remains unresolved with multiple competing views on how to approach the problem.

Contextual Notes

There are limitations in the discussion, such as missing assumptions about the properties of the roots and the specific relationships needed to describe the Galois group.

algekkk
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Anybody can help me show that Gal(E/Q) is isomorphic to Z4? E is the splitting field for X^5-1 over Q. Thanks.
 
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[itex]x^5- 1= (x- 1)(x^4+ x^3+ x^2+ x+ 1)[/itex] has the single real root, x= 1, and 4 complex roots, [itex]e^{2\pi i/5}[/itex], [itex]e^{4\pi i/5}[/itex], [itex]e^{6\pi i/5}[/itex], and [itex]e^{8\pi i/5}[/itex]. Can you construct the Galois group from that? What does Z4 look like?
 
Z4 is {0,1,2,3} I can tell that their orders are all four. Just not sure about what's the rest needed to show isomorphic.
 
What orders are all four? Only two elements of Z4 have an order of 4.

Do you know what a relationship between the non-trivial roots of x5-1 is that allows you to describe all the roots in terms of one of them?
 
Ok, thanks for the help. I have this one solved. All I need to do is to show the Galois group are cyclic.
 

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