Rings, Modules, Fields

13 Normal extensions

We would like to understand the automorphisms of a field extension \(L/K\), namely the automorphisms of \(L\) as a \(K\)-algebra, which we denote \(\Aut (L/K)\). Sometimes these don’t exist, as in the case of \(\QQ (2^{\frac {1}{3}})/\QQ \), but other times they do, such as in the case of \(\QQ (\sqrt {a})/\QQ \). When we try to see why this property doesn’t hold for \(\QQ (2^{\frac {1}{3}})\), we see it is because the conjugates of \(2^{\frac {1}{3}}\) are not contained in the field.

Definition 13.1. We say a polynomial over \(K\) splits in an extension \(L\) if it factors in \(L\) into linear terms.

Definition 13.2. An algebraic extension \(L/K\) is normal if every irreducible polynomial over \(K\) that has a root in \(L\) splits.

Definition 13.3. We call the splitting field of a collection of polynomials \(f \in K[x]\) the smallest normal field extension containing a root of \(f\).

Definition 13.4. We say that two embedded algebraic extensions \(L/K,\tilde {L}\) are conjugate if there is an automorphism in \(\Aut (\bar {K}/K)\) taking \(L\) to \(\bar {L}\).

From the extension lemma it follows that two extensions are isomorphic iff they are conjugate.

Proposition 13.5. TFAE:
- 1. \(L/K\) is normal.
- 2. If \(\alpha \in L\), its conjugates are in \(L\).
- 3. Every automorphism of \(\bar {K}/K\) restrict to automorphisms of \(L/K\).
- 4. \(L\) is the splitting field of some collection of polynomials.
- 5. \(L\) has only one conjugate.

Proof. The first two are clearly equivalent since roots correspond to points in the algebraic closure. If all the conjugates are in \(L\), since automorphisms of \(\bar {K}/K\) send conjugates to conjugates, they fix \(L\). The inverse also fixes \(L\), so it restricts to an automorphism of \(L\). If every automorphism sends \(L/K\) to itself, then by the extension lemma, \(L\) 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. □

Corollary 13.6 (Extension Lemma). If \(L/K\) is normal, any automorphism of \(K\) extends to one of \(L\).

Proof. It extends to the algebraic closure, and then restricts to \(L\). □

Note that since \(Aut(\bar {K}/K)\) acts transitively on conjugates and restricts to automorphism of any normal extension, normal extensions act transitively on the conjugates. Also note that purely inseparable extensions are normal.

Corollary 13.7. Splitting fields are unique up to isomorphism.

Proof. They can embed into the Galois closure in one way. □

The algebraic closure is the splitting field of every polynomial, and the separable closure is the splitting field of every separable polynomial.

Corollary 13.8. If \(M/L/K\) are extensions, \(M/K\) is normal implies \(M/L\) is normal. The compositum and intersection of normal extensions is normal.

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. □

Definition 13.9. The normal closure of an extension \(L/K\) is the smallest normal extension of \(K\) containing \(K\)

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.