Rings, Modules, Fields
13 Normal extensions
We would like to understand the automorphisms of a field extension
From the extension lemma it follows that two extensions are isomorphic iff they are conjugate.
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Proof. The first two are clearly equivalent since roots correspond to points in the algebraic closure. If all the conjugates are in
, since automorphisms of send conjugates to conjugates, they fix . The inverse also fixes , so it restricts to an automorphism of . If every automorphism sends to itself, then by the extension lemma, is normal. A normal extension is the splitting field of the minimal polynomials of all its elements, and a splitting field is by definition normal. The last statement is clearly equivalent to the third. □
Note that since
The algebraic closure is the splitting field of every polynomial, and the separable closure is the splitting field of every separable polynomial.
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Proof. The first statement is obvious from the third equivalent version of normal. The statement about composita follows from Lemma 12.10, and the one about intersections is obvious. □
By Corollary 12.10 it is obtained by adjoining the conjugates of generating elements of the extension. In particular, the normal closure of a finite extension is finite.