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Lecture 17 - Feb 16

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Algebraic Extensions and Separable Degree

Let be an algebraic extension.

Fix an embedding

into an algebraic closure .

Let

Then

Moreover, is independent of the choice of and .

Simple Extensions

Let , where is algebraic over .

Let be the minimal polynomial of over .

Then every extension is determined by the choice of a root of in .

Hence

Tower Formula (Separable Degree)

Let be fields, with algebraic.

Then

Sketch of Proof

Let be an embedding.

  • Let be the extensions of to .
  • For each , let be the extensions of to .

Then each is an extension of to .

Thus

Finite Extensions

Now suppose is finite.

Then

Definition

  1. An algebraic element over is separable if

  2. A polynomial over is separable if all its roots are distinct.

  3. An algebraic extension is separable if every is separable over .

Observation

If is separable over and is a root of in an algebraic closure, then the minimal polynomial of over divides .

Since has distinct roots, so does the minimal polynomial.

Hence is separable.

Stability Under Intermediate Fields

If

and is separable, then is separable.

Indeed, the minimal polynomial over divides the minimal polynomial over .

Proposition

Let be a field extension containing two field extensions and .

If is separable, then

is separable.

Sketch

Let be separable over .

Then its minimal polynomial over is separable.

Since separability is preserved under base change, remains separable over .

Hence is separable.

Theorem

Let be a finite extension.

Then the following are equivalent:

  1. is separable.
  2. .

Proof

Since is finite, write

If is separable, then each is separable.

Using the tower formula for separable degrees,

Conversely, if

then each simple extension in a tower must satisfy equality, hence each is separable.

Thus is separable.

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