Let $\zeta _ { 5 } \in \mathbb { C }$ be a primitive 5th root of unity; let $\sqrt [ 5 ] { 2 }$ denote a real 5th root of 2, and let $l$ denote a square root of $-1$. Let $K = \mathbb { Q } \left( \zeta _ { 5 } , \sqrt [ 5 ] { 2 } \right)$. (A) Find the degree $[ K : \mathbb { Q } ]$ of the field $K$ over $\mathbb { Q }$. (B) Determine if $l \in \mathbb { Q } \left( \zeta _ { 5 } \right)$. (Hint: You may use, without proof, the following fact: if $\zeta _ { 20 } \in \mathbb { C }$ is a primitive 20th root of unity, then $\left[ \mathbb { Q } \left( \zeta _ { 20 } \right) : \mathbb { Q } \right] > 4$.) (C) Determine if $l \in K$.
Let $\zeta _ { 5 } \in \mathbb { C }$ be a primitive 5th root of unity; let $\sqrt [ 5 ] { 2 }$ denote a real 5th root of 2, and let $l$ denote a square root of $-1$. Let $K = \mathbb { Q } \left( \zeta _ { 5 } , \sqrt [ 5 ] { 2 } \right)$.\\
(A) Find the degree $[ K : \mathbb { Q } ]$ of the field $K$ over $\mathbb { Q }$.\\
(B) Determine if $l \in \mathbb { Q } \left( \zeta _ { 5 } \right)$. (Hint: You may use, without proof, the following fact: if $\zeta _ { 20 } \in \mathbb { C }$ is a primitive 20th root of unity, then $\left[ \mathbb { Q } \left( \zeta _ { 20 } \right) : \mathbb { Q } \right] > 4$.)\\
(C) Determine if $l \in K$.