By Patrick Morandi

The aim of this e-book is twofold. First, it really is written to be a textbook for a graduate point direction on Galois thought or box thought. moment, it really is designed to be a reference for researchers who want to know box concept. The ebook is written on the point of scholars who've familiarity with the elemental suggestions of team, ring, vector area thought, together with the Sylow theorems, factorization in polynomial jewelry, and theorems approximately bases of vector areas. This booklet has a number of examples and workouts, a lot of issues lined, and whole proofs given for the acknowledged effects. to aid readers clutch box.

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**Example text**

For property 2, if KIF is purely inseparable, then each min(F, a) splits over K, since the only root of min( F, a) is a itself. 28. If (J E Gal(KI F), then, for any a E K, the automorphism (J maps a to a root of min(F, a). Thus, (J(a) = a, so (J = id. Therefore, Gal(KI F) = {id}. If [K : F] < 00, then I{ is finitely generated over Fj say, K = F(al,' .. 20 it suffices by illdlldioll U) prove this ill the case K = F(a). But then [K : F] = deg(min(F,a)), which is a power of p by the previous lemma.

This Galois group is Abelian and is isomorphic to 7l/271 x 'lL/271. The subgroups of C = Gal(K/Q) are (id) , (CJ) , (T) , (CJT) ,C. w w -+ -+ w, w2 . (Y6) , «J!. TCJ. The (id) , (CJ) , (T) , (o"T) , (0"2T) ,G. (\;/2), Q(w 2 \Y2), Q(w\;/2), Q. One way to verify that these 'fields are in fact the correct ones is to show that, for any of these fields, the field is indeed fixed by the appropriate subgroup and its dimension over «J! is correct. ( ij2) ~ F(T). Since the index [G : (T)] = 3, we must have (F(Y) : F] = 3.

Proof. This corollary follows immediately from the preccding corollary since any finite extension of a field of characteristic 0 is separable. 0 The normal closure of a field extension Let K be an algebraic extension of F. The normal closure of Kj F is the splitting field over F of the set {min(F, a) : a E I(} of minimal polynomials of clements of IC As we will show below, the llormal closnre N of the extension J( j F is a llliuilllalnol"lllal extellsion of jI' which contains J(. This is reasonable since, for each a E 1<, the polynomial min(F, a) splits over any normal extension of F containing K.