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Lecture 10 - Feb 4

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Finite Extension and Automorphism Group

Let be a finite extension. Then

Proposition

Let be a field and .

  1. If , define

Then is a subfield of .

  1. If , then
  1. If and , then

Hence

  1. If , then
  1. Suppose and there exists such that

Then

Correspondence (Subfields ↔ Subgroups)

There is a correspondence:

given by

In general, the inverse map need not be surjective unless the extension is Galois.

Example

Let

where is a primitive cube root of unity.

Then

Possible automorphisms:

  • and conjugation on :

Altogether 6 automorphisms, forming a group isomorphic to .

Let be a subgroup generated by some .

Then compute

Example

Is

a Galois extension?

Here

Any must fix and send to another root of

But the other two roots are not in .

Hence the only possibility is

Thus

So the extension is not Galois.

Proposition

If is finite, then

Definition (Character)

Let be a group.

A character of in a field is a group homomorphism

(Any distinct characters are linearly independent.)

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