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Mathematics · 术语表

什么是 Galois extension？

又称： Galois group

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

A finite extension $L/K$ is Galois if it is the splitting field of a separable polynomial over $K$ . Its Galois group is $\operatorname{Gal}(L/K) = \operatorname{Aut}_K(L)$ , the group of field automorphisms of $L$ fixing $K$ 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 $L$ of $X^3 - 2$ over $\Q$ ( $j = \eu^{2\iu\pi/3}$ ): subgroups of $\operatorname{Gal}(L/\Q) \cong S_3$ (left, order reversed) match intermediate fields (right). The unique normal proper subgroup $\langle(1\,2\,3)\rangle$ corresponds to the unique subextension $\Q(\iu\sqrt3)/\Q$ that is Galois; the three conjugate subgroups $\langle(i\,j)\rangle$ correspond to the three conjugate cubic fields $\Q(j^k\sqrt[3]2)$ , none of them normal over $\Q$ .

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