Galois Extension of Q isomorphic to Z/3Z

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In summary, constructing a Galois extension E of Q such that Gal[E,Q] is isomorphic to Z/3Z involves finding a cubic root of a number x that is not in Q and adjoining it to Q. However, this is not always true and there are more complicated cases, such as considering the splitting field of a cubic polynomial with a perfect square discriminant.
  • #1
AlbertEinstein
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Hi...

How do I construct a Galois extension E of Q(set of rational numbers) such that Gal[E,Q] is isomorphic to Z/3Z.

Thanks.
 
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  • #2
There is a very good discussion of this on the http://en.wikipedia.org/wiki/Inverse_Galois_problem" [Broken]
 
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  • #3
Let d=cubic root of some number x s.t. d is not in Q. Then [Q(d):Q]=3 => o(Gal(Q(d)/Q))=3 and only group of order 3 is Z/3Z so you have your desired group.
 
  • #4
ABarrios said:
Let d=cubic root of some number x s.t. d is not in Q. Then [Q(d):Q]=3 => o(Gal(Q(d)/Q))=3 and only group of order 3 is Z/3Z so you have your desired group.

This is not necessarily true. For example, adjoining a primitive cube root of unity to Q generates a field extension of order 2. It is a bit more complicated than that, unfortunately!
 
  • #5
mrbohn1 said:
This is not necessarily true. For example, adjoining a primitive cube root of unity to Q generates a field extension of order 2. It is a bit more complicated than that, unfortunately!

When I wrote the above I was thinking of x as a natural number or integer, in which case it should work as an example of a Galois group isomorphic to Z/3Z. But as for a complete generalization, I do not know of how to give it.
 
  • #6
Consider the splitting field of any cubic polynomial with a perfect square discriminant. The root of the discriminant is rational, so it is fixed by all the Galois automorphisms. This means that there are no 2-cycles in the Galois group. If the polynomial is irreducible, then its Galois group is a nontrivial subgroup of S_3, hence is Z/3Z by eliminating all other possibilities.
 

1. What is the Galois Extension of Q isomorphic to Z/3Z?

The Galois Extension of Q is a field extension of the rational numbers that contains all the roots of a given polynomial. Isomorphic to Z/3Z means that the structure of this field is equivalent to that of the set of integers modulo 3.

2. What is the significance of this isomorphism?

This isomorphism allows us to study the properties of the Galois Extension of Q by using the familiar structure of the integers modulo 3. This makes it easier to analyze and understand the properties of this field.

3. How is the Galois Extension of Q isomorphic to Z/3Z related to algebraic number theory?

The Galois Extension of Q isomorphic to Z/3Z is a fundamental concept in algebraic number theory. It helps us understand the behavior of polynomials and the roots of these polynomials in relation to the rational numbers.

4. Can you provide an example of a polynomial that has roots in the Galois Extension of Q isomorphic to Z/3Z?

One example is the polynomial x^3 - 2. This polynomial has three roots in the Galois Extension of Q isomorphic to Z/3Z, namely the cube roots of 2: ∛2, ∛2e^(2πi/3), and ∛2e^(4πi/3).

5. How is the Galois Extension of Q isomorphic to Z/3Z used in practical applications?

The Galois Extension of Q isomorphic to Z/3Z has many practical applications in fields such as cryptography, coding theory, and error correction. It is also used in the construction of finite fields, which have numerous applications in computer science and engineering.

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