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Lecture 01 - Jan 7

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Ring Homomorphisms and Characteristic

Let be a commutative ring with unity.

If , then

Any ring homomorphism satisfies

unless is the zero map (in which case it is not very interesting).

Suppose is a field. Define a map

  • is a ring homomorphism.
  • is an integral domain.
  • Hence is a prime ideal in .

Therefore,

Since every ideal in is of the form ,

where either or for some prime .

We define the characteristic of as:

  • if ,
  • if prime.

Prime Subfield

Definition (Prime Subfield):

Let be a field. The smallest subfield of is called the prime subfield of .

Case 1:

Consider

If , then .

Hence extends to a morphism

defined by

If is any subfield of , then

Identifying with its image in , we obtain

Thus is the prime subfield.

Case 2:

Again consider

Now

Hence

is a subfield of .

Therefore the prime subfield of is .

Field Extension

Definition (Field Extension):

Let be a field. A field is called a field extension of if

We denote this by .

Viewing as a vector space over , the degree of the extension is defined as

Examples

  1. , ,
    All have characteristic .

  2. Let

    Then is a field and

    A basis of over is

    Hence

Finite Fields

Suppose is a finite field.

Consider

Then

so

Thus is a vector space over .

Since is finite,

Let

Then

Hence every finite field has order for some prime .

Tower Law

Suppose

Then is also an extension of and:

if both degrees are finite.

  1. If either or is infinite, then is infinite.

Proof of Multiplicativity

Assume

Let

be a basis of over .

Let

be a basis of over .

Take . Then

Since each ,

Hence

Thus the set

spans over .

To check linear independence, suppose

Rewriting,

Since are linearly independent over ,

Since are linearly independent over ,

Therefore the set has elements and forms a basis of over .

Hence

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