Lecture 05 - Jan 16

Composite Field

Let and be field extensions.

The composite field of and over is denoted by

Let

Then

Define

Then

the field of fractions of .

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Elements of the Composite Field

Any element is of the form

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Proposition

If is algebraic, then

is algebraic.

Similarly, if is algebraic, then

is algebraic.

In particular, if both and are algebraic, then

is algebraic.

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Proof Sketch

It is enough to prove that for any and ,

is algebraic over .

Since and is algebraic,

Because ,

is algebraic over .

Also, , hence algebraic over .

Therefore,

is algebraic over .

Hence every element of is algebraic over .

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Finite Extensions and Composite Fields

Suppose and are finite extensions.

Then:

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Proof Idea

Since and are finite, there exist

Then

Hence

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Divisibility

Because

we get

Similarly,

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Upper Bound

Let

Then by the Tower Law,

We claim

Indeed, writing

we get

Thus

Therefore,

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Example 1

Let

Then

Since

we see

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Example 2

Let

where is a primitive cube root of unity.

Then

Then

We have

Since

we compute

Thus