Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3 / d

Table of contents
- Solution d

Solution

(Z /7 Z)^{\times}

is cyclic of order six, generated by

3

. Put

ω = e^{2 πi /6}

. The six characters are

χ_{j} (3^{r}) = ω^{j r}, 0 \leq j \leq 5,

and vanish on multiples of

7

. Since

(3^{0}, 3^{1}, \dots, 3^{5}) \equiv (1, 3, 2, 6, 4, 5) (mod 7),

their values are explicitly

a χ_{j} (a) 11 2 ω^{2 j} 3 ω^{j} 4 ω^{4 j} 5 ω^{5 j} 6 ω^{3 j} .

Taking

j = 0, \dots, 5

lists all six characters.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version