StudyLabs

Lecture 18 - Feb 18

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Proposition

Let be an algebraic extension.

An element is separable over if and only if its minimal polynomial
has distinct roots.

Let

Then

If is algebraic, define

Proposition

Let be a finite extension. Then the following are equivalent:

  1. is separable over .
  2. .

Proof

(i) (ii)

Since is finite, write

Using multiplicativity of separable degrees,

Since each is separable,

Hence

(ii) (i)

Assume

Let .

Using the tower law,

Also,

Since equality holds globally, we must have

Thus is separable. Hence is separable.

Proposition

Let be a field containing extensions and .

If is separable, then the compositum is separable.

Sketch

Let

Then is the field generated by .

If is separable over , it remains separable over .

Hence elements of are separable over , so is separable.

Derivatives and Separability

Let

Define the formal derivative:

Properties:

Proposition

Let .

  1. is a multiple root of
    if and only if

  2. is separable
    if and only if

Proof (Sketch)

If is a multiple root, then

Differentiating,

Hence divides , so .

Conversely, if and , then divides .

Thus is separable and have no common non-constant factor.

Irreducible Case

Let be irreducible of degree .

Since :

  • If , then .
  • Hence an irreducible polynomial is separable
    if and only if
    .

Characteristic

If :

For any irreducible non-constant ,

Hence:

Every irreducible polynomial over a field of characteristic is separable.

Therefore all finite extensions of characteristic fields are separable.

Remark (Characteristic )

The above is false in characteristic .

Example

Let

Then

Thus is irreducible but not separable.

Frobenius Map

If , define the Frobenius map:

This is a field homomorphism (not necessarily surjective).

Polynomials of the form

have zero derivative and are inseparable.

Conclusion

  • Over fields of characteristic : all algebraic extensions are separable.
  • Over fields of characteristic : inseparable extensions can occur.
  • Separability is controlled by the formal derivative and the condition

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