Lecture 11 - Feb 5

Finite Extensions and Characters

If is finite, then

---

Characters

Let be a group.

A character of in a field is a group homomorphism

---

Examples

1. If is a field homomorphism between fields, take

   then

   is a character.

2. Let be a field extension and algebraic over .

   Let be the minimal polynomial of over .

   Assume contains all the roots of , say

   Define embeddings

   Then restricting to units,

   gives characters.

---

Proposition

Let be a group.

If are distinct characters of into a field , then they are linearly independent over .

---

Proof

Suppose

with not all , and minimal.

Since , there exists such that

For all ,

Multiply by and compare with evaluation at :

Subtracting suitably yields a shorter nontrivial linear relation among

contradicting minimality.

Hence the characters are linearly independent.

---

Proposition

If is a finite extension, then

---

Proof

Let

Let be a -basis of .

Consider the matrix

If , then the rows are linearly dependent over .

So there exist , not all zero, such that

Since is a basis, this implies

as functions .

Restricting to , this gives a nontrivial linear relation among distinct characters

contradicting the linear independence of characters.

Therefore

i.e.,