Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by
\( x^{4} - 10x^{2} + 20 \)
| $\times$ | \(\chi_{ 40 } ( 1, ·)\) | \(\chi_{ 40 } ( 3, ·)\) | \(\chi_{ 40 } ( 27, ·)\) | \(\chi_{ 40 } ( 9, ·)\) |
|---|---|---|---|---|
| \(\chi_{ 40 }(1, ·)\) | \(\chi_{ 40 } ( 1, ·)\) | \(\chi_{ 40 } ( 3, ·)\) | \(\chi_{ 40 } ( 27, ·)\) | \(\chi_{ 40 } ( 9, ·)\) |
| \(\chi_{ 40 }(3, ·)\) | \(\chi_{ 40 } ( 3, ·)\) | \(\chi_{ 40 } ( 9, ·)\) | \(\chi_{ 40 } ( 1, ·)\) | \(\chi_{ 40 } ( 27, ·)\) |
| \(\chi_{ 40 }(27, ·)\) | \(\chi_{ 40 } ( 27, ·)\) | \(\chi_{ 40 } ( 1, ·)\) | \(\chi_{ 40 } ( 9, ·)\) | \(\chi_{ 40 } ( 3, ·)\) |
| \(\chi_{ 40 }(9, ·)\) | \(\chi_{ 40 } ( 9, ·)\) | \(\chi_{ 40 } ( 27, ·)\) | \(\chi_{ 40 } ( 3, ·)\) | \(\chi_{ 40 } ( 1, ·)\) |