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Lecture 02 - Jan 9

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Tower Law (continued)

Suppose

Adjoining Elements

Let be a field extension and .

  • The smallest subring of containing and is denoted by

  • The smallest subfield of containing and is denoted by

Proposition

Proof (Sketch)

  • The right-hand side of (1) is a ring.
  • The right-hand side of (2) is the field of fractions of the ring in (1).

Let be any ring containing and .

Since , every monomial in lies in .

Because , we get

Hence the description is correct.

For (2), take the field of fractions of (1).
The proof is analogous.

Generated Subfield

The field

is called the subfield of generated by over .

Finitely Generated Extensions

  • Every finite extension is finitely generated.
  • A finitely generated extension need not be finite.

A finitely generated extension is of the form

We say is finitely generated if

for some .

Simple Extensions

An extension is called a simple extension if

for some .

Proposition

If , then

Proof (Idea)

Since

we have

Also

Thus

Similarly, the reverse inclusion holds.

More generally,

Constructing Field Extensions with a Root

Let be a field and .

We want a field extension of containing a root of .

Take a maximal ideal of containing .

Then

is a field.

Let

be the natural projection and set

Since , we get

Thus is a field extension of .

Now,

So is a root of in .

Algebraic Construction via a Root

Suppose is irreducible over and is a root.

Define

Then

Hence

So

If is another root of , then

(They are isomorphic as extensions of , though not necessarily the same subfield.)

If

and , then

Moreover,

Example

The numbers and are two roots of

where is a root of

Then

(They are isomorphic as field extensions.)

Graph View