Math Is Fun Forum

ASADM

Member

Registered: 2009-03-05

Posts: 18

Normal Closure....

Hi guys,

I need to find the Normal Closures of:

a) A = Q(Sqrt(-5+2*Sqrt(5)))

b) B=  Q(Sqrt(1+Sqrt(5)))

both over Q.

I really don't know what it is you're supposed to do. (our lecturer is really unclear! )

I think it's something to doing with saying "this is a subset of that is a subset of...etc" & conjugates.

If someone could please explain I would be extremely greatful! :-)

Many thanks. x

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Normal Closure....

Recall that the normal closure of a field L is the smallest field containing L that is normal.  Recall that a field K is normal over F provided it is the splitting field of some collection C of polynomials in F[x].

Your task is to first find the minimal polynomials for each of these numbers, and then find all the roots of such polynomials.  The field generated by these roots will be normal (since it is the splitting field of the minimal polynomial).  It is also the smallest normal field containing a root of the minimal polynomial (why?).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."