Eleventh cyclotomic field ($\mathbb{Q}({\zeta }_{11})$)

The Galois group of $\mathbb{Q}({\zeta }_{11})/\mathbb{Q}$ isIts only subfield degrees over $\mathbb{Q}$ are $1,2,5,10$, and every subfield is Galois because this group is abelian.

The unique quadratic subfield of $\mathbb{Q}({\zeta }_{11})$ is

The fixed field of complex conjugation in $\mathbb{Q}({\zeta }_{11})$ is the degree-five field $\mathbb{Q}({\zeta }_{11}+{\zeta }_{11}^{-1})$.