Stanley Yao Xiao

D_4-quartic fields with monogenic cubic resolvent

A number field L over \mathbb{Q} of degree 4 is said to be a D_4-field if the Galois closure M of L has Galois group isomorphic to the dihedral group D_4 of order 8. A quartic number field L is said to have monogenic cubic resolvent if its ring of integers \mathcal{O}_L has a monogenic cubic resolvent ring. In this talk, we shall count the number of quartic fields with monogenic cubic resolvent sorted by conductor, defined to be the Artin conductor of the unique irreducible 2-dimensional Galois representation $\rho_M : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\mathbb{C})$ which factors through \text{Gal}(M/\mathbb{Q}).