 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 123

The group $(\mathbb Z/7\mathbb Z)^\times$ is cyclic of order six, generated by $3$. Put $\omega=e^{2\pi i/6}$. The six characters are
$$
\chi_j(3^r)=\omega^{jr},
\qquad 0\leq j\leq5,
$$
and vanish on multiples of $7$. Since
$$
(3^0,3^1,\ldots,3^5)\equiv(1,3,2,6,4,5)\pmod7,
$$
their values are explicitly
$$
\begin{array}{c|rrrrrr}
a&1&2&3&4&5&6\\ \hline
\chi_j(a)&1&\omega^{2j}&\omega^j&\omega^{4j}&\omega^{5j}&\omega^{3j}.
\end{array}
$$
Taking $j=0,\ldots,5$ lists all six characters.


Solution

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)