Group table for the character group for $\textrm{Gal}(K/\mathbb{Q})$

$K$ is the global number field defined by \(x^{2} \) \(\mathstrut +\mathstrut 1 \)

$\times$ \(\chi_{ 4 } ( 1, ·)\) \(\chi_{ 4 } ( 3, ·)\)
\(\chi_{ 4 }(1, ·)\) \(\chi_{ 4 } ( 1, ·)\) \(\chi_{ 4 } ( 3, ·)\)
\(\chi_{ 4 }(3, ·)\) \(\chi_{ 4 } ( 3, ·)\) \(\chi_{ 4 } ( 1, ·)\)