Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$
$K$ is the global number field defined by \(x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)
| $\times$ | \(\chi_{ 5 } ( 1, ·)\) | \(\chi_{ 5 } ( 2, ·)\) | \(\chi_{ 5 } ( 3, ·)\) | \(\chi_{ 5 } ( 4, ·)\) |
|---|---|---|---|---|
| \(\chi_{ 5 }(1, ·)\) | \(\chi_{ 5 } ( 1, ·)\) | \(\chi_{ 5 } ( 2, ·)\) | \(\chi_{ 5 } ( 3, ·)\) | \(\chi_{ 5 } ( 4, ·)\) |
| \(\chi_{ 5 }(2, ·)\) | \(\chi_{ 5 } ( 2, ·)\) | \(\chi_{ 5 } ( 4, ·)\) | \(\chi_{ 5 } ( 1, ·)\) | \(\chi_{ 5 } ( 3, ·)\) |
| \(\chi_{ 5 }(3, ·)\) | \(\chi_{ 5 } ( 3, ·)\) | \(\chi_{ 5 } ( 1, ·)\) | \(\chi_{ 5 } ( 4, ·)\) | \(\chi_{ 5 } ( 2, ·)\) |
| \(\chi_{ 5 }(4, ·)\) | \(\chi_{ 5 } ( 4, ·)\) | \(\chi_{ 5 } ( 3, ·)\) | \(\chi_{ 5 } ( 2, ·)\) | \(\chi_{ 5 } ( 1, ·)\) |