What is the Galoisgroup of the field extension of complex numbers over rationals?

Answer 1

The Galois group of the field extension of the complex numbers (\mathbb{C}) over the rational numbers (\mathbb{Q}) is trivial, which means it consists only of the identity element. This is because (\mathbb{C}) is an algebraically closed field, and any nontrivial field automorphism of (\mathbb{C}) would have to fix (\mathbb{Q}) while also permuting roots of polynomials. However, since the only roots of polynomials with coefficients in (\mathbb{Q}) that can exist in (\mathbb{C}) are the roots of unity and these cannot be permuted without affecting the field structure, the only automorphism is the identity. Thus, the Galois group is trivial, denoted as ({ \text{id} }).