Let L be the Galois closure of the finite extension Q(a) of Q. From any prime p dividing the order of Gal(L/Q) prove there is a subfield F of L with [L:F] = p and L = F(a).
2006-12-20 07:06:08 · 6 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
If G=Gal(L:F) and you can use a group theory result to "extract an order p" (Sylow theroems? that was along time ago!), i.e, find H
2006-12-20 08:32:29
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answer #1
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answered by a_math_guy 5
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In mathematics, more specifically in abstract algebra, Galois theory, named after Ãvariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.
Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.
Further abstraction of Galois theory is achieved by the theory of Galois connections.
2006-12-20 07:07:45 · answer #2 · answered by Anonymous · 0⤊ 1⤋
In Herstein's Topics in Algebra, there is a proof for when a prime divides the order of a group, that group must necessarily have a subgroup of order p (result by Cauchy). There might be some similarities in the proofs if you look it up.
2006-12-20 08:27:46 · answer #3 · answered by Professor Maddie 4 · 0⤊ 0⤋
Think about the properties of a Galois group, what is significant about the prime? Here you will find your answer...
2006-12-20 07:20:25 · answer #4 · answered by Topologist 1 · 0⤊ 0⤋
www.nrich.maths.org/public/viewer.php?obj_id=1422&part=index&refpage=monthindex.php -
2006-12-20 07:34:35 · answer #5 · answered by raj 7 · 0⤊ 0⤋
WHOZ HE?
2006-12-20 07:13:12 · answer #6 · answered by Kat W 2 · 0⤊ 2⤋