Completed Partially
Hi, I am going to indulge myself in mathematical banter...
1) If you recall I posted some thoughts on the ideas to solve my Field Theory tutorials. Seems that I was of on the wrong track to begin with. The problem of determining the F-isomorphism of a extension F(u_1,....,u_r) can be reduced to the case of the simple extension F(u_1) -> F(v_1). Obviously the isomorphism is totally determined by the image u_1 under the mapping, which must be its conjugate. We can then 'move one level higher', by extending the isomorphism to F(u_1,u_2), which contains F(u_1) . Again, the mapping is determined fully by the image of u_2 and so on.
So in general we can map F(u_1,...u_i) -> F(v_1,....,v_i) recursively like this. F-> F is the identity map.
sigma_r ( sum[ a_i*u^i]) = sum[ sigma_r-1(a_i) * v^i]. It is easy to see now that the F-isomorphism is totally determined by the images of u_1...u_r alone.
2) The part on where an F-automorphism permuted the roots of the minimal polynomial is quite easy to see, but I don't really know how to put it down in words. Shall continue to think about it.
3) I proved that the fixed field of a subset, S of Aut (E) is the same that of its generated subgroup. That's easy to show, but a totally non-trivial observation.
4) I still don't know how to determine all the F-automorphisms of a transcendental extension.
and....
5) I can't show that the real numbers only has one automorphism..i.e. the identity.
1) If you recall I posted some thoughts on the ideas to solve my Field Theory tutorials. Seems that I was of on the wrong track to begin with. The problem of determining the F-isomorphism of a extension F(u_1,....,u_r) can be reduced to the case of the simple extension F(u_1) -> F(v_1). Obviously the isomorphism is totally determined by the image u_1 under the mapping, which must be its conjugate. We can then 'move one level higher', by extending the isomorphism to F(u_1,u_2), which contains F(u_1) . Again, the mapping is determined fully by the image of u_2 and so on.
So in general we can map F(u_1,...u_i) -> F(v_1,....,v_i) recursively like this. F-> F is the identity map.
sigma_r ( sum[ a_i*u^i]) = sum[ sigma_r-1(a_i) * v^i]. It is easy to see now that the F-isomorphism is totally determined by the images of u_1...u_r alone.
2) The part on where an F-automorphism permuted the roots of the minimal polynomial is quite easy to see, but I don't really know how to put it down in words. Shall continue to think about it.
3) I proved that the fixed field of a subset, S of Aut (E) is the same that of its generated subgroup. That's easy to show, but a totally non-trivial observation.
4) I still don't know how to determine all the F-automorphisms of a transcendental extension.
and....
5) I can't show that the real numbers only has one automorphism..i.e. the identity.
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on Thursday, February 24, 2005 at 3:34 PM.
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10:48 PMhello...er...how come i can't see ur tagboard? anyway, just to let you know, im still following your blog ardently. you must be the single most regular blogger ive ever seen! not to mention thought-provoking. mind if i link your site on mine? =^) daniel
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