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Summary: A direct constructive treatment of the Fundamental Theorem
of Galois Theory
DRAFT
Peter Aczel
September 29, 1998
The Fundamental Theorem of Galois Theory
This is the basic structure theorem at the heart of Galois Theory. We start with a classical
account of the result and then explain how this needs to be modified to get a constructive
result. We will restrict attention to fields of characteristic 0.
Let E be a field (of characteristic 0). We write AutE for the group of automorphisms
of E. The fundamental theorem concerns correspondences each way between the subfields
of E and the subgroups of AutE . To each subfield F of E is associated the subgroup F 4
of AutE given by
F 4 = fg 2 AutE j 8x 2 F g(x) = xg:
When we want to make explicit the dependence of F 4 on E we will write it Gal(E : F )
and call it the Galois group of E : F . To each subgroup G of AutE is associated the
subfield G r of E given by
G r = fx 2 E j 8g 2 G g(x) = xg:
These correspondences form a Galois connection; i.e. for each F and each G
F ` G r () G ` F 4 :
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