Lecture 03 - Jan 12

Algebraic Extensions

Definition

Let be a field extension and let .

We say is algebraic over if there exists a nonzero polynomial

such that

If is not algebraic over , then is called transcendental over .

An extension is called algebraic if every is algebraic over .

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Algebraic Element and Evaluation Map

Suppose is a field extension and is algebraic over .

Define

Since is algebraic, .

Let

If satisfies , then

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Minimal Polynomial

Among all nonzero polynomials such that , choose the monic polynomial of least degree.

Definition

The monic polynomial of least degree

such that

is called the minimal polynomial of over .

Its degree is called the degree of over .

Moreover,

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Examples

(1)

Let and consider

Then satisfies

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(2) Primitive th Root of Unity

Let

Then satisfies

Also,

Hence satisfies

The minimal polynomial of has degree (in this case).

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Degree Relations in Tower

Suppose

Then

By the Tower Law,

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Example with Nested Extensions

Let

Then

Over :

Over :

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Finite Degree Implies Algebraic

If

then is algebraic over .

Conversely, if is algebraic over , then

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Two Algebraic Elements

Suppose are algebraic over .

Then

Since

by the Tower Law,

Because is algebraic over , it is algebraic over , so

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Converse

If

then both and are algebraic over .

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Characterisation

Let be a field extension and .

Then

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Example

Let

Then

Since

and

we get

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Algebraic Closure Properties

Let be a field extension.

1. If are algebraic over , then

   are algebraic over .

2. The set of all algebraic elements in over forms a subfield of .

3. Every finite extension of is algebraic.