Lecture 12 - Feb 6

Let

Then

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Assume

Let

be a -basis of .

Consider the matrix

If , then the rows are linearly dependent over .

So there exist , not all zero, such that

Equivalently,

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Claim

as maps .

Indeed, let

Then

Thus the are linearly dependent as functions .

Restricting to

this gives a nontrivial linear relation among distinct characters

contradicting linear independence of characters.

Hence

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Theorem (Equality Case)

Let be a field and

be finite.

Define

Then

and

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Proof Sketch

Note that

Hence

Let .

Assume

Then .

Choose

linearly independent over .

Consider the matrix

Since , the columns are linearly dependent.

Take a minimal linear dependence

Applying any gives another relation.

Subtracting yields a shorter dependence unless all .

This contradicts linear independence over .

Hence

Thus

Finally,

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Example: Finite Fields

Let

be a finite field with elements.

Let

be the polynomial ring over .

Let

be its quotient field.

Define the Frobenius map

This map is injective but not surjective in general.