Normal extension of rational complex fields
Solution 1:
Any time you adjoin one $n$-th root of an element downstairs, your extension will be normal if the downstairs field contains all the $n$-th roots of unity. That’s the case here, with $n=5$.
Any time you adjoin one $n$-th root of an element downstairs, your extension will be normal if the downstairs field contains all the $n$-th roots of unity. That’s the case here, with $n=5$.