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themadhatter012@yahoo.com science forum beginner
Joined: 28 Sep 2005
Posts: 24

Posted: Wed Jun 21, 2006 6:41 pm Post subject:
Nth Roots of Unity And Groups



I'm trying to work problems from a book but 2 have me stumped because
the book does not define what the "nth roots of unity" are  besides
that it has something to do with the complex numbers  so I have no
idea how to do the problems.
The first is to let H be the subset of C (complex numbers) consisting o
fthe nth roots of unity. Prove that H is a group under multiplication.
The second is let G be the multiplicative group of the nth roots of
unity. Prove that G is isomorphic to (Z/(n), +).
Any help on the concept and problems would be appreciated.
TMH 

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Stephen J. Herschkorn science forum Guru
Joined: 24 Mar 2005
Posts: 641

Posted: Wed Jun 21, 2006 7:11 pm Post subject:
Re: Nth Roots of Unity And Groups



themadhatter012@yahoo.com wrote:
Quote:  I'm trying to work problems from a book but 2 have me stumped because
the book does not define what the "nth roots of unity" are  besides
that it has something to do with the complex numbers  so I have no
idea how to do the problems.
The first is to let H be the subset of C (complex numbers) consisting o
fthe nth roots of unity. Prove that H is a group under multiplication.
The second is let G be the multiplicative group of the nth roots of
unity. Prove that G is isomorphic to (Z/(n), +).
Any help on the concept and problems would be appreciated.

The nth roots of unity are the complex numbers cos (2 k pi / n) + i
sin (2 k pi / n) for integer k.
You really should have a grasp on the basic algebra of complex numbers
before reading anything at all advanced such as group theory. Any high
school algebra text should do.

Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan 

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Shmuel (Seymour J.) Metz1 science forum Guru
Joined: 03 May 2005
Posts: 604

Posted: Wed Jun 21, 2006 11:18 pm Post subject:
Re: Nth Roots of Unity And Groups



In <1150915281.594792.89370@r2g2000cwb.googlegroups.com>, on
06/21/2006
at 11:41 AM, themadhatter012@yahoo.com said:
Quote:  I'm trying to work problems from a book but 2 have me stumped because
the book does not define what the "nth roots of unity" are

Exactly what the name suggests; z is an nth root of unity if z^n=1.
Quote:  besides that it has something to do with the complex numbers

While that is generally assumed, you can discuss the nth roots of
unity in other fields.

Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk Email subject to legal action. I reserve the
right to publicly post or ridicule any abusive Email. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org 

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hale@tulane.edu science forum beginner
Joined: 23 Apr 2006
Posts: 13

Posted: Wed Jun 21, 2006 11:19 pm Post subject:
Re: Nth Roots of Unity And Groups



themadhatter012@yahoo.com wrote:
Quote:  I'm trying to work problems from a book but 2 have me stumped because
the book does not define what the "nth roots of unity" are  besides
that it has something to do with the complex numbers  so I have no
idea how to do the problems.

The complex number w is an nth root of unity if w^n = 1.
Let C be the field of complex numbers.
The set of nth roots of unity = { w in C: w^n = 1}
Quote: 
The first is to let H be the subset of C (complex numbers) consisting o
fthe nth roots of unity. Prove that H is a group under multiplication.

Thus, H = { w in C: w^n = 1}.
It is easy to show that H is a group under multiplication,
assuming that the set of complex numbers C is a field.
Quote:  The second is let G be the multiplicative group of the nth roots of
unity. Prove that G is isomorphic to (Z/(n), +).

Thus, G = { w in C: w^n = 1}.
You need to analyse G.
For example, you probably will need to show that G has
exactly n elements.
 Bill Hale 

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Virgil science forum Guru
Joined: 24 Mar 2005
Posts: 5536

Posted: Wed Jun 21, 2006 11:21 pm Post subject:
Re: Nth Roots of Unity And Groups



In article <449999FA.8020804@netscape.net>,
"Stephen J. Herschkorn" <sjherschko@netscape.net> wrote:
Quote:  themadhatter012@yahoo.com wrote:
I'm trying to work problems from a book but 2 have me stumped because
the book does not define what the "nth roots of unity" are  besides
that it has something to do with the complex numbers  so I have no
idea how to do the problems.
The first is to let H be the subset of C (complex numbers) consisting o
fthe nth roots of unity. Prove that H is a group under multiplication.
The second is let G be the multiplicative group of the nth roots of
unity. Prove that G is isomorphic to (Z/(n), +).
Any help on the concept and problems would be appreciated.
The nth roots of unity are the complex numbers cos (2 k pi / n) + i
sin (2 k pi / n) for integer k.
You really should have a grasp on the basic algebra of complex numbers
before reading anything at all advanced such as group theory. Any high
school algebra text should do.

The definition of an nth root of unity is any number x such that
x^n = 1, so that 1 is always such a root.
If one allows complex numbers there will always be exactly n distinct
such nth roots, and it will turn out that the product of any two of them
will also be one of them, and the sum of all of them will always be zero.
n Set of roots (where i^2 = 1)
 
1 { 1 }
2 { 1, 1 }
3 { 1, (1+i*sqrt(3))/2, (1i*sqrt(3))/2 }
4 { 1, 1, i , i }
and so on 

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