T2. 9. Dirichlet Characters

Question 1

Define a Dirichlet character

Answer 1

A homomorphism χ from G → C* where C* is the multiplicative group of non-zero complex numbers

Question 2

Give the homomorphism condition on the characters

Answer 2

χ(ab) = χ(a)χ(b)

Question 3

Define the principle character

Answer 3

The character χ_0(a) = 1 for all a in G

Question 4

Define χ_1χ_2(a)

Answer 4

= χ_1(a)χ_2(a)

Question 5

Define the row orthogonality for characters here

Answer 5

SUM_g∈G χ(g) = { #G if χ = χ0
{ 0 else

Question 6

Define column orthogonality for characters here

Answer 6

SUM_χ ∈G~ χ(g) = { #G if g = e_G
{ 0 else

Question 7

Define a Dirichlet character (words)

Answer 7

A character over the multiplicative group: i.e. multiplication of a set of numbers mod q

Question 8

Define a Dirichlet character (formula)

What is the trivial character?

Answer 8

χ(n) = {χ(n modq) if (n,q) = 1
{0 else

Under this condition, the trivial character remains 1

Question 9

How many Dirichlet characters modq?

Answer 9

φ(q) = #{a ∈ N, a ≤ n and (a,n) = 1}

Question 10

Give the row orthogonality for Dirichlet characters modq

Answer 10

SUM χ(n) = {φ(q) if χ = χ_0
{ 0 else

for (n,q) = 1
n runs from 0 to q

Question 11

Give the column orthogonality for Dirichlet characters modq

Answer 11

SUM χ(n) = {φ(q) if n = 1modq
{ 0 else

for χ modq

Question 12

SUM_ χ modq χ*(a)χ(b) = ??

Answer 12

{φ(q) if n = a modq
{0 else