Part (ii) or (ii)' of the Fundamental Theorem (2.5) is equivalent to: an intermediate field E is
Part (ii) or (ii)' of the Fundamental Theorem (2.5) is equivalent to: an intermediate field E is normal over K if and only if the corresponding subgroup E' is normal in G = AutKF in which case Gi E' ≅ AutKE. [See Exercise 18.]
Data from exercise 18
Let F be normal over K and E an intermediate field. Then E is normal over K if and only if E is stable [see Exercise 17]. Furthermore AutKF/E' ≅ AutKE
Data from Exercise 17
If an intermediate field Eis normal over K, then E is stable (relative to F and K).
Data from theorem 2.5
(Fundamental Theorem of Galois Theory) If F is a finite dimensional Galois extension of K, then there is a one-to-one correspondence between the set of all 1A Galois extension is frequently required to be finite dimensional or at least algebraic and is defined in terms of normality and separability, which will be discussed in Section 3. In the finite dimensional case our definition is equivalent to the usual one. Our definition is essentially due to Art in, except that he calls such an extension "normal." Since this use of "normal" conflicts (in case char F ≠ 0) with the definition of "normal" used by many other authors, we have chosen to follow Artin·s basic approach, but to retain the (more or less) conventional terminology intermediate fields of the extension and the set of all subgroups of the Galois group AutKF (given by E|→E' = AutEF) such that:
(i) the relative dimension of two intermediate fields is equal to the relative index of the corresponding subgroups; in particular, AutKF has order [F: KJ;
(ii) F is Galois over every intermediate field E, but E is Galois over K if and only if the corresponding subgroup E' = AutEF is normal in G = AutKF; in this case G/E' is (isomorphic to) the Galois group AutKE of E over K. The proof of the theorem (which begins on p. 251) requires some rather lengthy preliminaries. The rest of this section is devoted to developing these. We leave the problem of constructing Galois extension fields and the case of algebraic Galois extensions of arbitrary dimension for the next section. The reader should note that many of the propositions to be proved now apply to the general case. As indicated in the statement of the Fundamental Theorem, the so-called Galois correspondence is given by assigning to each intermediate field E the Galois group AutEF of F over E. It will turn out that the inverse of this one-to-one correspondence is given by assigning to each subgroup H of the Galois group its fixed field in F. It will be very convenient to use the ·•prime notation" of Theorem 2.3, so that E' denotes AutEF and H' denotes the fixed field of H in F. It may be helpful to visualize these priming operations schematically as follows. Let L and M be intermediate fields of the extension K ⊂ F and let J,H be subgroups of the Galois group G = AutKF.
Formally, the basic facts about the priming operations are given by
Data from Theorem 2.3
Let F be an extension field of K, E an intermediate field and Ha subgroup of AutKF. Then
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