Normal Extensions and Field Automorphisms

An algebraic extension L/K is normal when every irreducible polynomial in K[x] with a root in L splits completely in L. The quantifier concerns irreducible polynomials over K; demanding that every polynomial over K split would be a much stronger claim. For example, Q(sqrt(2)) is normal over Q because it is the splitting field of x squared minus two. It certainly does not contain roots of every rational polynomial.

A finite extension is normal precisely when it is a splitting field of some polynomial over its base. For the forward direction, choose finitely many algebraic generators and multiply their minimal polynomials: normality supplies every conjugate, while the chosen generators already generate the field. Conversely, a base-field embedding permutes the roots of a splitting polynomial, so it preserves the whole field they generate. This embedding viewpoint is often easier to use than testing infinitely many polynomials.

A K-automorphism of L is a bijective field homomorphism from L to itself that fixes each element of K. Its value on a generator must be another root of the generator's minimal polynomial. This is a necessary condition, but several generators can satisfy additional relations, so their images cannot always be chosen independently. The safe procedure is to present the field by generators and relations, then verify that each proposed assignment respects every relation.

The real field Q(cube root of two) illustrates the missing distinction. Its generator has three distinct complex conjugates, yet only one is real. Any Q-automorphism must therefore fix the generator, so the automorphism group is trivial even though the extension has degree three. Its other Q-embeddings land in different subfields of the complex numbers. Adjoining a primitive cube root of unity supplies the missing roots and produces a larger normal extension.

A finite Galois extension is both normal and separable. These requirements do different jobs: separability ensures enough distinct embeddings, and normality keeps their images inside one field. Together they imply that the automorphism group has size equal to the extension degree. Characteristic zero makes separability automatic, which is why it can disappear from rational-number examples without disappearing from the theorem itself.

In characteristic p, the extension F_p(t)(t to the power one over p) is normal because x to the p minus t has only one distinct root and already splits in that extension. It is inseparable of degree p, and its base-fixing automorphism group is trivial. Thus a count of symmetries is also a diagnostic: too few automorphisms may signal missing conjugates, repeated conjugates, or both. Identify which hypothesis failed before applying a Galois theorem.