Formation of a magnetosphere for gas giants / small stars
We do not know whether Jupiter has a solid core. That is one of the primary goals of the Juno mission.

In general it is clear that a solid core is not required to generate magnetic fields. The Sun does so and magnetic fields have been measured (with average surface field strengths of around 3 kG) on very low mass stars. Incidentally, the Sun's magnetic field does have a large scale dipole component.

More recently, there have been observations of cyclotron and auroral emission suggesting that even old, cold brown dwarfs have strong, organised magnetic fields. See for example Kuznetsov et al. (2012); Hallinan et al.(2015) and references therein.

http://arxiv.org/abs/1111.7019
http://arxiv.org/abs/1507.08739

At the moment, nobody can say for sure how exactly these fields are generated, sustained and organised.

In fact you could do worse than look at a paper that popped up at the top of today's astro-ph listings by Williams et al.  The abstract begins:

"The well-studied M9 dwarf TVLM 513-46546 is a rapid rotator (P_rot ~ 2 hr) hosting a stable, dipolar magnetic field of ~3 kG surface strength."

http://arxiv.org/abs/1511.05559