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Galois theory Introduction. - math.ou.edu

Basic idea of Galois theory is to study fieldextensions by relating them to their automorphism groups. Recall thatanF-automorphism ofE/Fis defined as an automorphism :E Ethat fixesFpointwise, that is, (a) =afor alla F. TheF-automorphisms ofE/Fform a group under composition (you can thinkof this as a subgroup ofS(E)). We call this theGalois groupofEoverFand denote it byGal(E/F) ={ :E E: is anF-automorphism}.Now consider an intermediate fieldF L E; I ll writeE/L/Ftorefer to this situation, but should issue a warning that this notation isnon-standard. Then we can similarly consider theL-automorphismsGal(E/L) ={ :E E: automorphism, (a) =afor alla L}.This is a subgroup of Gal(E/F) since any such in particular leavesF Linvariant. Conversely, if we are given a subgroupH Gal(E/F),then we can introduceInv(H) ={a E: (a) =afor all H}.

Galois theory 6.1. Introduction. The basic idea of Galois theory is to study eld extensions by relating them to their automorphism groups. Recall that an F-automorphism of E=F is de ned as an automorphism ’: E! E that xes F pointwise, that is, ’(a) = afor all a2F. The F-

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Linear Algebra 3rd edition http://joshua.smcvt.edu ...

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RUDIMENTARY GALOIS THEORY, Galois Theory, Theory, AN INTRODUCTION TO GALOIS THEORY, INFINITE GALOIS THEORY, Galois, Outline of Galois Theory Development, GALOIS THEORY FOR ARBITRARY FIELD EXTENSIONS