Showing Gal(E/Q) is Isomorphic to Z4

  • Thread starter algekkk
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In summary, the conversation is about proving that the Galois group Gal(E/Q) is isomorphic to Z4. E is the splitting field for X^5-1 over Q, with one real root and four complex roots. The Galois group can be constructed from the four complex roots and is shown to be cyclic. Z4 is described as the set {0,1,2,3} with two elements having an order of 4. The relationship between the non-trivial roots of x^5-1 allows for all the roots to be described in terms of one of them.
  • #1
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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  • #2
[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?
 
  • #3
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.
 
  • #4
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?
 
  • #5
Ok, thanks for the help. I have this one solved. All I need to do is to show the Galois group are cyclic.
 

FAQ: Showing Gal(E/Q) is Isomorphic to Z4

What is Gal(E/Q)?

Gal(E/Q) stands for the Galois group of the extension E/Q. It is a group of automorphisms that preserve the field structure of the extension E/Q.

What does it mean for Gal(E/Q) to be isomorphic to Z4?

If two groups are isomorphic, it means that they have the same structure and can be mapped onto each other. In this case, Gal(E/Q) and Z4 have the same structure and can be represented by the same group of elements.

Why is it important to show that Gal(E/Q) is isomorphic to Z4?

This is important because it helps us understand the properties of the extension E/Q. Isomorphism allows us to compare two groups and determine if they have similar characteristics. In this case, it tells us that Gal(E/Q) has the same properties as Z4, which can help us in further analysis and calculations.

How is the isomorphism between Gal(E/Q) and Z4 established?

The isomorphism between Gal(E/Q) and Z4 is established by constructing a one-to-one correspondence between the elements of the two groups. This is done by finding a suitable mapping between the elements of Gal(E/Q) and Z4 that preserves the group structure.

What are some applications of this isomorphism?

This isomorphism has several applications in mathematics, especially in the field of Galois theory. It allows us to study the properties of Gal(E/Q) by using the known properties of Z4. It also helps in solving equations and understanding the behavior of polynomials in the extension E/Q.

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