cyclotomic_extension
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From: Splittingfield on 27 Apr 2010 00:28 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. 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)? Thanks.
From: William Elliot on 27 Apr 2010 04:50 On Tue, 27 Apr 2010, Splittingfield wrote: > Let x be a primitive 13th root of unity over Q. > Let y be a primitive 7th root of unity over Q. > Let i^13 = 1 = j^7 be the primitive roots. i = exp 2pi.i/13 j = exp 2pi.i/7 k = exp 2pi.i/91 = exp 2pi.6i/7 * exp 2pi.(-11)/13 = j^6 * i^(-11) = j^6 * i^2 k^91 = 1 > I know [Q:Q(x)]=12 and [Q:Q(y)]=6, but I don't think [Q(x,y):Q] = 72. > What is the value of [Q(x,y):Q]? 90 > Is it the same with Gal(Q(x,y)/Q)? > If not, what is value of Gal(Q(x,y)/Q)? I don't know.
From: Derek Holt on 27 Apr 2010 05:11 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.
From: Splittingfield on 27 Apr 2010 01:37 > 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)? Thanks a lot.
From: Gerry Myerson on 27 Apr 2010 19:30
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) |