Hi guys,
How do you find the normal closure Q(Sqrt(1+Sqrt(5))) over Q?
I have absolutely no idea how to do this! :-s I think it has something to do with chains of subfields & conjugate, but if some could please explain the general method & how to apply it to this question I would be extremely greatful! :-)
Many thanks in advance! x
Ok, so does that means the normal closure is:
Q(Sqrt1+Sqrt(5)),Sqrt(1-Sqrt(5))) then?
& could you please exaplin why this is the case? The definition for normal closure that I have is that it is the smallest extension of the current extension which is normal.
Many thanks. :-)
Let . This polynomial is also irreducible.
The zeros of this polynomial are: .
Therefore, the normal closure is:
Notice that .
Thus, it is also okay to write: .
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Here is an explanation of why we take the normal closure to be splitting field. First we define what we mean by "normal closure". Let be an algebraic field extension, the normal closure of over is the splitting field of over . The idea is that the normal closure is the "smallest" normal extension over containing . By "smallest" we mean that if is a normal extension over which satisfies then . To prove this let and consider , since is normal it immediately means that splits over and so contains all the roots of . But is the field generated by the roots of this polynomials, hence, which forces . As a consequence we can prove that the normal closure of is the splitting field over for each . Let be this splitting field, then obviously with normal, therefore it forces . Your problem is a special case when , so the normal closure is the splitting field of over .
Fantastic! :-D
Thank you SO much theperfecthacker! :-)
In an question online I've seen a chain of fields like:
Q SUBFIELD OF Q(Sqrt(3)) SUBFIELD OF Q(Sqrt(1+2Sqrt(3)),Sqrt(1-2Sqrt(3))).
Is this true in general?
ie -
Q SUBFIELD OF Q(Sqrt(A)) SUBFIELD OF Q(Sqrt(a+bSqrt(A)), Sqrt(a-bSqrt(A))) ?
If so, this means that
Q(Sqrt(A))(Sqrt(a+bSqrt(A)),Sqrt(a-bSqrt(A)
= Q(Sqrt(a+bSqrt(A)),Sqrt(a-bSqrt(A)) , yes?
Furthermore, is Q(Sqrt(A)) the ONLY quadratic subfield of Q(Sqrt(a+bSqrt(A)),Sqrt(a-bSqrt(A))) ? If not, what are the others? Indeed, what are all the subfields of Q(Sqrt(a+bSqrt(A)),Sqrt(a-bSqrt(A))) ?
Many thanks. :-)