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Nth Roots of Unity And Groups
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Wed Jun 21, 2006 11:21 pm    Post subject: Re: Nth Roots of Unity And Groups Reply with quote

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, (-1-i*sqrt(3))/2 }
4 { 1, -1, i , -i }
and so on
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hale@tulane.edu
science forum beginner


Joined: 23 Apr 2006
Posts: 13

PostPosted: Wed Jun 21, 2006 11:19 pm    Post subject: Re: Nth Roots of Unity And Groups Reply with quote

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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Shmuel (Seymour J.) Metz1
science forum Guru


Joined: 03 May 2005
Posts: 604

PostPosted: Wed Jun 21, 2006 11:18 pm    Post subject: Re: Nth Roots of Unity And Groups Reply with quote

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 E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org
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Stephen J. Herschkorn
science forum Guru


Joined: 24 Mar 2005
Posts: 641

PostPosted: Wed Jun 21, 2006 7:11 pm    Post subject: Re: Nth Roots of Unity And Groups Reply with quote

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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themadhatter012@yahoo.com
science forum beginner


Joined: 28 Sep 2005
Posts: 24

PostPosted: Wed Jun 21, 2006 6:41 pm    Post subject: Nth Roots of Unity And Groups Reply with 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.


TMH
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