Galois Groups of Extensions by Roots of Unity

In summary: It seems that this is a much more complicated question than just looking at the values of n that work, as there are many possible situations in which a unit group might be cyclic.
  • #1
lugita15
1,554
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Consider field extensions of the form Q(u) where Q is the field of rational numbers and [itex]u=e^{\frac{2\pi i}{n}}[/itex], the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else?

Any help would be greatly appreciated.

Thank You in Advance.
 
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  • #2
read these notes starting from page 46.

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]
 
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  • #3
mathwonk said:
read these notes starting from page 46.

http://www.math.uga.edu/%7Eroy/843-2.pdf [Broken]
Thanks mathwonk! So now the question becomes, for what n is the unit group of Zn cyclic?
 
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  • #4
lugita15 said:
Consider field extensions of the form Q(u) where Q is the field of rational numbers and [itex]u=e^{\frac{2\pi i}{n}}[/itex], the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else?

Any help would be greatly appreciated.

Thank You in Advance.



As [itex]Gal\left(\mathbb Q(u)/\mathbb Q\right) \cong \left(\mathbb Z/n\mathbb Z\right)^{*}[/itex] , the question is the same as asking what multiplicative

groups of units modulo n are cyclic...

They are precisely when n is: a prime, a power of AN ODD prime and twice a prime. you can check this in most decent group theory books, and

any of these groups is of order [itex]\phi(n)[/itex]

DonAntonio
 
  • #5
DonAntonio said:
As [itex]Gal\left(\mathbb Q(u)/\mathbb Q\right) \cong \left(\mathbb Z/n\mathbb Z\right)^{*}[/itex] , the question is the same as asking what multiplicative

groups of units modulo n are cyclic...

They are precisely when n is: a prime, a power of AN ODD prime and twice a prime. you can check this in most decent group theory books, and

any of these groups is of order [itex]\phi(n)[/itex]

DonAntonio
DonAntonio, where would I find the proof that unit group of Zn is cyclic if n is the power of an odd prime, and that the values of n you mentioned are the only ones that work?
 
  • #6
I'm also curious to know, under what circumstances is the unit group of a finite ring cyclic, in general?
 
  • #7
read the following notes, pages 48-50.

http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]
 
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  • #8
mathwonk said:
read the following notes, pages 48-50.

http://www.math.uga.edu/%7Eroy/844-2.pdf [Broken]
Thanks, I had forgotten that you can apply the structure theorem for abelian groups, since Zn* is obviously abelian. Do you have any thoughts on my other question, namely what are the circumstances in general for a unit group of a finite ring to be cyclic?
 
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What is a Galois group of an extension by roots of unity?

A Galois group of an extension by roots of unity is a group that describes the automorphisms of the extension field that fix the base field and act on the roots of unity. It is a fundamental concept in Galois theory and allows for the study of algebraic extensions and their symmetries.

What is the significance of studying Galois groups of extensions by roots of unity?

Studying Galois groups of extensions by roots of unity allows for a deeper understanding of the structure and properties of algebraic extensions. It also has applications in fields such as number theory, cryptography, and algebraic geometry.

How is the order of a Galois group related to the degree of the extension?

The order of a Galois group is equal to the degree of the extension. This is a result of the Galois correspondence, which states that there is a one-to-one correspondence between subgroups of the Galois group and intermediate fields of the extension. Therefore, the number of subgroups (and thus the order of the group) is equal to the number of intermediate fields, which is equal to the degree of the extension.

Can a Galois group of an extension by roots of unity be non-abelian?

Yes, a Galois group of an extension by roots of unity can be non-abelian. This means that the group is not commutative, and the order of applying automorphisms matters. In fact, the Galois group of an extension by roots of unity is non-abelian if and only if the extension is not a cyclotomic extension.

What is the relationship between the Galois group and the solvability of an algebraic equation?

The Galois group of an extension by roots of unity plays a crucial role in determining the solvability of an algebraic equation. If the Galois group is solvable, then the corresponding equation is solvable by radicals. On the other hand, if the Galois group is not solvable, then the equation is not solvable by radicals. This is known as the Galois solvability criterion.

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