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From: Splittingfield on 27 Apr 2010 16:46
> In article
> <1768273977.29136.1272361099241.JavaMail.root(a)gallium.
> mathforum.org>,
> Splittingfield <K(a)k.com> wrote:
>
> > > On 27 Apr, 09:28, Splittingfield <K...(a)k.com>
> wrote:
> > > > Let x be a primitive 13th root of unity over Q.
> > > > Let y be a primitive 7th root of unity over Q.
> > > >
> > > > What is the value of [Q(x,y):Q]?
> > > >
> > > > I know [Q:Q(x)]=12 and [Q:Q(y)]=6, but I don't
> > > think [Q(x,y):Q] = 72.
> > >
> > > Why do you not think that? Q[x,y] is just the
> > > cyclotomic field of 91-
> > > th roots of unity, and there are exactly 72 such
> > > primitive roots.
> > >
> > > > What is the value of [Q(x,y):Q]? Is it the same
> > > with Gal(Q(x,y)/Q)? If not, what is value of
> > > Gal(Q(x,y)/Q)?
> > >
> > >
> > > All cyclotomic fields are Galois extensions of Q,
> > > because if you have
> > > one n-th primitive root of unity in the field,
> then
> > > you have them all.
> > >
> > > Derek Holt.
> >
> > If gcd(x, y) =/=1, then how do I find [Q(x,y):Q]
> and Gal(Q(x,y)/Q)?
> >
> > For instance, x is the primitive 18th root of unity
> and y is the primitive
> > 12th root of unity. Then [Q(x,y):Q]= eulerpi( lcm
> (12,
> > 18))=eulerpi(36)=12=Gal(Q(x,y)/Q)?
>
> I think you've got it.
>
> If x is a primitive m-th root and y a primitive n-th
> root
> then each is a p-th root where p = LCM(m, n)
> so Q(x, y) is contained in Q(z) where z is a
> primitive p-th root.
> But also z is in Q(x, y) by elementary number theory
> so Q(x, y) = Q(z).
>
> But you write some funny things. When you write
> gcd(x, y),
> you really mean gcd(order of x, order of y). When you
> write
> Gal(Q(x, y) / Q) = 12, you really mean the order of
> Gal(Q(x, y) / Q)
> is 12. You might be less confused if you wrote what
> you actually
> mean, instead of writing something sorta kinda like
> what you mean.
>
> --
> Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for
> email)

Right.

Thank you for your correction.
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