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Mathematics · Glossary

What is Galois extension?

Also known as: Galois group

Definition 4.18 University Mathematics — Year 3 · Chapter 4 — Field Extensions and Galois Theory

A finite extension L/KL/K is Galois if it is the splitting field of a separable polynomial over KK. Its Galois group is Gal⁡(L/K)=Aut⁡K(L)\operatorname{Gal}(L/K) = \operatorname{Aut}_K(L), the group of field automorphisms of LL fixing KK pointwise.

The Galois correspondence for the splitting field L of X3 - 2 over ℚ (j = 2 π/3): subgroups of Gal(L/ℚ) S_3 (left, order reversed) match intermediate fields (right). The unique normal proper subgroup (1\,2\,3) corresponds to the unique subextension ℚ( √3)/ℚ that is Galois; the three conjugate subgroups (i\,j) correspond to the three conjugate cubic fields ℚ(jk√[3]2), none of them normal over ℚ.
The Galois correspondence for the splitting field LL of X3−2X^3 - 2 over Q\Q (j=e2iπ/3j = \eu^{2\iu\pi/3}): subgroups of Gal⁡(L/Q)≅S3\operatorname{Gal}(L/\Q) \cong S_3 (left, order reversed) match intermediate fields (right). The unique normal proper subgroup ⟨(1 2 3)⟩\langle(1\,2\,3)\rangle corresponds to the unique subextension Q(i3)/Q\Q(\iu\sqrt3)/\Q that is Galois; the three conjugate subgroups ⟨(i j)⟩\langle(i\,j)\rangle correspond to the three conjugate cubic fields Q(jk23)\Q(j^k\sqrt[3]2), none of them normal over Q\Q.
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