The discussion centers on the Galois nature of field extensions involving \(\mathbb{Q}(\sqrt{-5})\) and \(\mathbb{Q}(\beta + i\beta)\), where \(\beta\) is the positive fourth root of 5. It is established that \(\mathbb{Q}(\sqrt{-5})\) is a Galois extension over \(\mathbb{Q}\) since it is the splitting field of the polynomial \(x^2 + 5\) with distinct roots. However, the Galois status of \(\mathbb{Q}(\beta + i\beta)\) over \(\mathbb{Q}\) and \(\mathbb{Q}(\sqrt{-5})\) remains uncertain, with attempts to identify explicit separable polynomials proving challenging.
PREREQUISITESStudents of abstract algebra, mathematicians specializing in field theory, and anyone interested in the properties of Galois extensions and polynomial roots.
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.