The discussion revolves around determining whether certain field extensions involving the fourth root of 5 and imaginary units are Galois over the rational numbers. The original poster presents specific extensions: \(\mathbb{Q}(\sqrt{-5})\), \(\mathbb{Q}(\beta + i\beta)\), and their relationships to each other and to \(\mathbb{Q}\).
Some participants have provided insights regarding the Galois nature of \(\mathbb{Q}(\sqrt{-5})\) and its polynomial representation. However, there remains uncertainty about the other extensions, with participants expressing difficulty in identifying explicit separable polynomials and questioning their Galois status.
Participants note challenges in identifying the relationships between the fields and the polynomials that would demonstrate their Galois properties. There is an acknowledgment of the complexity involved in proving or disproving the Galois nature of the extensions under discussion.
shoplifter said:also, thinking about the polynomial (x + \beta + i\beta)(x + \beta - i\beta)(x - \beta + i\beta)(x - \beta - i\beta) doesn't help, because it gives me the information that \mathbb{Q}(\beta + i\beta) is galois over \mathbb{Q}(\sqrt{5}), but nothing else, and i don't see how this helps in the problem.