Lecture 07 - Jan 29

Finite Extensions and Automorphisms

Let be a finite extension.

This means is a finite dimensional vector space over , i.e.,

We ask:

• What is ?
• What is ?
• Is finite?

---

Determination by Generators

Suppose

Any is completely determined by the images of the generators:

Thus possible automorphisms correspond to choices

provided the map extends to a field automorphism fixing .

---

Induced Action on Polynomials

Let .

Then

Define an induced map

by

If , then all coefficients lie in , hence

so

---

Images of Algebraic Elements

Let be algebraic over , and let

be its minimal polynomial.

Then

Applying ,

Thus

is also a root of .

Hence, for each ,

must be one of the finitely many roots of its minimal polynomial.

Therefore,

is finite.

---

Example 1

Let

The minimal polynomial is

whose roots are

Thus there are two automorphisms:

• Identity:
• Conjugation:

Hence

---

Example 2

Let

The minimal polynomial is

Its roots in are

where is a primitive cube root of unity.

But only

lies in .

Thus the only possible image is itself, so

---

Example 3

Let

where is a primitive cube root of unity.

The roots of are

Possible images of are these three roots.

Possible images of are

Thus there are possibilities.

In fact,

---

General Question

Given a field , can we find a field containing such that every polynomial over has a root in ?

(This leads to the notion of algebraic closure.)